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Doctoral Thesis
Embedding problems in symplectic geometry
Author(s):
Schlenk, Felix
Publication Date:
2001
Permanent Link:
https://doi.org/10.3929/ethz-a-004232481
Rights / License:
In Copyright - Non-Commercial Use Permitted
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ETH Library
Diss. ETH No. 14254
Embedding problems
in
symplectic geometry
A dissertation submitted to the
EIDGENÖSSISCHE TECHNISCHE
for the
HOCHSCHULE
of
degree
Doctor of Mathematics
presented by
FELIX SCHLENK
Dipl.
born
Math. ETH Zürich
September 15,
on
citizen of Zürich and
accepted
on
1970
Kilchberg
the recommendation of
Prof. Eduard
Zehnder, examiner
Prof. Dietmar Salamon, co-examiner
Zürich 2001
ZÜRICH
Contents
Abstract
i
Zusammenfassung
iii
Acknowledgements
v
Introduction
1
Rigidity
1.1
1.2
1.3
2
3
1
2
5
9
Reformulation of Theorem 2
2.2
The
2.3
End of the
9
folding
proof
construction
3.1
Modification of the
3.2
Multiple folding
Embeddings into
Embeddings into
Symplectic folding
in
18
26
Multiple symplectic folding
in four dimensions
31
construction
31
folding
32
balls
37
cubes
54
higher
4.1
Four
4.2
Embedding polydiscs
Embedding ellipsoids
4.3
6
Comparison
Rigidity for ellipsoids
Rigidity for poly discs?
2.1
3.4
5
1
of the relations <,•
Proof of Theorem 2
3.3
4
vii
dimensions
types of folding
65
into cubes
67
into balls
74
Proof of Theorem 3
95
5.1
Proof of lim
p*{M,
5.2
Proof of lim
p%(M,
5.3
Asymptotic embedding
Symplectic
6.1
6.2
versus
co)
=
1
95
co)
=
1
112
invariants
140
Lagrangian folding
Lagrangian folding
Comparison
of
65
symplectic
143
143
and
Lagrangian folding
152
7
Proof of Theorem 4
159
Appendix
173
A
Proof of
B
Symplectic capacities
B.l
173
and the invariants c# and cq
Intrinsic and extrinsic
R2"
B.4
B.5
The invariant ce
B.3
D
1
symplectic capacities
Symplectic capacities on R2
Comparison of Cß and cc with symplectic capacities
Comparison of Cß with the symplectic diameter
B.2
C
Proposition
as a
symplectic projection
The Extension after Restriction
Computer
programs
on
Principle
invariant
179
179
182
187
188
189
193
217
D.l
The estimate seb
217
D.2
The estimate sec
219
References
221
Lebenslauf
225
Abstract
Fix
In -dimensional
a
dard
symplectic
manifold
symplectic
space
(R2ra, coq),
(M, co) and
open subset U of stan¬
an
where
(i>o
^2 d*i
=
A
àyl.
1=1
The
symplectic embedding problem associated with U and (M, co)
largest a > 0 for which there exists a symplectic embedding of aU
U} into (M, co). We choose U to be an open symplectic ellipsoid
"
E{rx,...,rn)
P(n,...,r„)
Here,
•
•
z
e
I2
polydisc
or a
•
\7
{az \
z„)eC"
(zi
=
asks for the
=
<
B2n{r)
52(ri)x-.-x52(rn).
denotes the open ball of radius
rn. We first choose
Theorem 1
=
Assume
of the ellipsoid E{r\,
r2
(M, co)
<
...
2
to
be
a
We may
r.
assume
that r\
<
ri
<
ball.
r2. T/zcn f/iere Joe^
,rn) into the ball
nof cxwf a
B2n(R) if R
<
symplectic embedding
rn.
This
non-embedding result is sharp and therefore sheds some light on the power
capacities, which are used in its proof. Indeed, we have the
following embedding result.
of Ekeland-Hofer
Theorem 2
Assume
E(n ,...,n,rn)
The basic
into
r2
>
2r2.
Then there exists
B2n(R)for all ^r2/2
+
r2
<
a
R
symplectic embedding of
<
rn.
ingredient in the proofs of all our embedding results is a refinement
of a symplectic folding method first used by F. Lalonde and D. McDuff. In par¬
ticular, Theorem 2 is proved by folding an ellipsoid once. It can be substantially
improved by folding more often. We in particular show that a ball can be asymp¬
totically filled with skinny ellipsoids. More generally, assume that (M, co) is a
11
connected
r
=
-^ fM con.
Vol(tt£(l,...,l,r))
pr(M, co)
sup
=
————
Vol
a
where the supremum is taken
cally
embeds into
Theorem 3
a
for which
aE2n(\,
...,
1, r) symplecti-
(M, co).
lim,--^ pr(M, co)
analogous
all
over
(M, CO)
result holds for
=
1.
polydiscs.
finally use the symplectic folding method to answer a question
plectic embeddings of balls into the standard symplectic cylinder
We
Z
By
Gromov's
into Z if
r
>
52(1)
=
x
R2"-2
=
inf sup
<P
where <p varies
over
all
area
(R2",
expectations
we
c{r)
co0).
Dx
0/or a/Z
r
e
]0, 1 [.
=
n
Dx)
/
B2n(r) into
B2{\) x {x}, x
symplectic embeddings
have
=
((p(B2n(r))
\
x
Z denotes the 2-dimensional open disc
Theorem 4
c
about sym¬
Nonsqueezing Theorem there is no symplectic embedding of B2n (r)
1. So assume r < 1, and consider the symplectic invariant
c(r)
to
For
1 set
>
An
(M2n, co)
manifold of finite volume Vol
symplectic
of
Z and where
Dx
c
R2"-2. Contrary
Zusammenfassung
Sei
(M, co) eine 2n-dimensionale symplektische Mannigfaltigkeit, und sei U eine
offene
Teilmenge
des
symplektischen
Raumes
(R2", coq), wobei
n
coq
=
^ dxt
A
dyt.
i=\
Das U und
grössten
(M, co) zugeordnete symplektische Einbettungsproblem fragt nach der
Zahl
a
>
0 für welche die
Menge
all
=
{az \
e
z
U] symplektisch
in
(M, co) eingebettet werden kann. Wir wählen U als ein offenes symplektisches
Ellipsoid
E(n,...,rn)
=
Ki I
z„)gC"
(zi
EJTT<1
r<
,=1
oder als eine
Polyscheibe
P(r1,...,rn)
Hier bezeichnet
B2n(r)
dass r\
•
Satz 1
<
r2
<
Sei
r2
lipsoids E(r\,
•
<
...,
<
2
=
B2(r1)x---xB2(rn).
den offenen Ball
vom
Radius
rn. Wir betrachten zuerst den
r2.
r„)
Dann existiert keine
in den Ball
B2n(R) falls
Wir können
annehmen,
Fall, dass (M, co) ein Ball ist.
r.
symplektische Einbettung
R
<
des El¬
rn.
Dieser
Nichteinbettungssatz ist scharf und zeugt daher von der Stärke der EkelandKapazitäten, die in seinem Beweis verwendet werden. Es gilt nämlich der
folgende Einbettungssatz.
Hofer
Satz 2
Sei
r2
lipsoids E(r\,
...
>
2r2.
,r\,rn)
Dann existiert eine
in
B2n (R) für alle
symplektische Einbettung
Jr2/2 + r2
<
R
<
des El¬
rn.
Die
grundlegende Technik in den Beweisen aller unserer Einbettungssätze ist eine
Verfeinerung einer symplektischen Faltungsmethode, die zuerst von F. Lalonde
and D. McDuff verwendet wurde.
ein
Ellipsoid
einmal falten.
Wir beweisen Satz 2 denn auch indem wir
Satz 2 kann durch mehrfaches Falten wesentlich
IV
verbessert werden. Wir
zeigen insbesondere,
dass ein Ball
asymptotisch mit dün¬
nen Ellipsoiden gefüllt werden kann. Wir betrachten allgemeiner eine zusammen¬
hängende 2«-dimensionale symplektische Mannigfaltigkeit (M, co) von endlichem
Volumen Vol(M, co)
^ fM con. Für r > 1 definieren wir
=
pr(M, CO)
Vol(ttg(l,...,l,r))
SUp
=
......
Vol
a
(M, CO)
wobei das
Supremum über all jene a genommen
symplektisch in (M, co) eingebettet werden kann.
Satz 3
linv-ôo pr(M, co)
Ein ähnlicher Satz
gilt
für
=
wird für die
aE2n(\,...,
1, r)
1.
Polyscheiben.
Wir verwenden die
über
symplektische Faltungsmethode schliesslich, um eine Frage
symplektische Einbettungen von Bällen in den symplektischen Zylinder
Z
=
B2{\) xR2""2
c
(R2n,co0)
beantworten.
Aufgrund von Gromovs Nichteinbettungssatz existiert keine sym¬
plektische Einbettung von B2n(r) in Z falls r > 1. Wir nehmen daher r < 1 an
und betrachten die symplektische Invariante
zu
c(r)
=
inf sup
<P
wobei cp über alle
area
Satz 4
den
c(r)
Erwartungen gilt
=
Ofür alle
r e
]0, 1 [.
Dx
=
n
Dx)
/
symplektischen Einbettungen
Z die 2-dimensionale offene Scheibe
Entgegen
(<p(B2n(r))
V
x
von
52(1)
x
B2n{r)
{x},
x
in Z läuft und
e
R2n~2,
Dx
C
bezeichnet.
Acknowledgements
It is
a
great pleasure
to express my
Zehnder for his support, his
Giving me
for
gratitude
patience,
the freedom of pursuing my
to my
advisor Professor Eduard
and his continuous interest in my work.
own
research interests, he made it
me to find my way as a mathematician.
His
possible
insight
prevented me more than once from further pursuing a wrong idea.
always found an encouraging word, and he never lost his humor, even not in
in mathematics and his
criticism
times.
Last but not
finally influenced,
I
was
very
co-referee my
hope, my own style.
happy when Professor Dietmar Salamon spontaneously agreed to
thesis. He helped to create a great atmosphere in the symplectic
of motivation for
Hofer,
I
me
source
Ilmanen, François Laudenbach, Thomas Mautsch, Dusa McDuff and
am
symplectic folding,
in
a
particular indebted to Dusa McDuff
technique basic for the whole thesis.
Discussions with Yuri Chekanov, Janko Latchev and Yuli
they
continuous
ideas of this work grew out of discussions with David Hermann, Helmut
Tom
results.
a
me.
Leonid Polterovich. I
to
bad
least, his great skill in presenting mathematical results has
group of ETH, and his enthusiasm for mathematics has been
Many
He
Including
them into this thesis would have made it
who
explained
Rudyak led to other
even longer, and so
will appear, I
hope, somewhere else.
Doing symplectic geometry at ETH has been greatly facilitated through the
existence of the symplectic group. When I started this thesis in 1996, this group
consisted of Casim Abbas, Michel Andenmatten, Kai Cieliebak,
Hansjörg Geiges,
Helmut
Hofer, Markus Kriener, Torsten Linnemann, Laurent Moatty, Matthias
Schwarz, Karl Friedrich Siburg, Edi Zehnder and myself, and when I finished my
thesis, the group consisted of Meike Akveld, Urs Frauenfelder, Ralph Gautschi,
Janko Latchev, Thomas Mautsch, Dietmar Salamon, Joa Weber, Katrin Wehrheim,
Edi Zehnder and
myself.
This thesis is the visible fruit of my studies in mathematics. A
fruit
are
the
friendships
with Rolf Heeb, Laurent
more
Ivo Stalder.
Sana, Selin kedim,
sonsuz
sabir
ve
important
Lazzarini, Christian Rüede and
sevgin için te§ekkür ederim.
Seite Leer /
Blank leaf
Introduction
Consider
is
a
a
connected smooth rc-dimensional manifold M. A volume form
smooth nowhere
vanishing
fM Œ is positive, and we write Vol (M, Q)
(not necessarily connected) subset U of R" with the
n-form £l.
orient M such that
each open
=
fM Q.
M
on
It follows that M is orientable.
We
We endow
Euclidean volume
form
Q.0
A smooth
embedding
cp
U
:
°^-
dx\
=
A
(U, Œo)
<
Vol
•
•
/\dxn.
M is called volume
<p*Q
Then Vol
•
preserving
if
£2o-
=
(M, Q). The following proposition shows that this ob¬
vious condition for the existence of
volume
a
preserving embedding
is the
only
one.
Proposition
ding
A
if and
1 The set U embeds into M
Vol
only if
proof of this
result
(U, Œo)
can
<
symplectic manifold (M, co)
non-degenerate closed 2-form co.
volume form, and that M is
open subset U of
R2n
is
a
smooth volume
Appendix
a
preserving embed¬
A.
smooth manifold M endowed with
The
even
by
(M, Œ).
be found in
A
a
Vol
non-degeneracy
dimensional,
with the standard
of
dim M
symplectic
co
=
implies
a
that
smooth
^con
is
2n. We endow each
form
n
coo
=
^ dxi
A
dy,.
i=ï
A smooth
embedding
cp
U
:
<^->
M is called
cp*co
In
particular,
and
À
>
—ycon
0
behind
every
induced
we set
Proposition
=
(M, co)l
the volume forms
Qo
=
-7^0
symplectic forms. Given an open subset U of R2" and
{kz 6 R2" \z e U}. In the symplectic world, the question
the
1 becomes
Problem 1 What is the
beds into
if
COQ.
symplectic embedding preserves
by
XU
=
symplectic
largest
number k such that
(kU, coo) symplectically
em¬
Introduction
Vlll
In dimension
embedding is volume preserving if and only if it is sym¬
plectic, and so Problem 1 is completely solved by Proposition 1. In higher dimen¬
sions, however, strong symplectic rigidity phenomena appear. Denote the open
2n-dimensional ball of radius r by B2n{jir2) and the open 2n-dimensional sym¬
plectic cylinder B2(a) x R2n_2 by
2,
an
Z2n(a)
1.
Examples
[15]
For
arbitrarily
x
R2n~2.
(Gromov's Nonsqueezing Theorem [12]) For«
symplectically
2.
B2(a)
=
n
embeds into the
>
cylinder Z2n(A) only
if A
> a.
2, there exist bounded starshaped domains U
small volume but do not
On the other hand, the
lem 1 becomes less
rigid
symplectically embed
following
two results
2, the ball B2n (a)
>
C
R2"
which have
B2n(jt).
into
suggest that the situation in Prob¬
if U is "thin". We denote
by
(zi,...,zn)en
E(ai, ...,an)
i=\
the open
symplectic ellipsoid in R2"
with radii
*Jat /n.
([13, p. 335] and [10, p. 579]) Consider a 2«-dimensional symplec¬
tic manifold (M, co). For any a > 0 there exists a (possibly very small!) e > 0
Examples
3.
such that the
4.
ellipsoid E2n(e,
...,
(Traynor, [34, Theorem 6.4])
e,
a) symplectically embeds into M.
For all k
>
1 and
>
0 there exists
a
symplectic
embedding
E(-^-,k7t\
is
Examples
interesting:
2 and 4 show that
already
Problem 2 What is the smallest ball
B\n
-+
the
+
).
following special
B2n(A)
into which U
case
of Problem 1
symplectically
em¬
beds?
In this work
we
investigate the
zone
of transition between
ity
in Problems 1 and 2. The main tool of
be
special symplectic invariants,
the
so
rigidity
detecting embedding
called
and flexibil¬
obstructions will
symplectic capacities (see [18]
and
Introduction
ix
Chapter 1). Unfortunately, symplectic capacities can be computed only for very
special sets. Therefore, we look at a model situation in which the set U is a sym¬
an). Since a permutation of the symplectic coordinate
plectic ellipsoid E{a\,
<
planes is a (linear) symplectic map, we may assume a\ < 02 <
an. We first
discuss our answers to Problem 2. Our main rigidity result is
...,
•
Theorem 1 Assume an
embedding
Our
of the
proof
Theorem 1
ellipsoid E{a\,
...,
an)
not
exist
symplectic
smooth
a
B2n(A)
into the ball
•
if A
<
an.
the n'th Ekeland-Hofer
uses
proved
was
Then there does
2a\.
<
•
in
[10]
as
capacity. In the special case n
application of symplectic homology.
an
=
2,
The
argument given here is much simpler and works in all dimensions.
Our first
embedding
result states that Theorem 1 is
Theorem 2 Assume an
ding of
>
ellipsoid E(a\,
the
2a\.
Then there exists
,a\,
...
a
into the ball
an)
sharp.
smooth
symplectic embed¬
B2n(an
8) for every 8 e
—
]o,af-ai[.
The reader
a
might ask why
we assume
an-\
=
a\ in
Theorem 2. This is because
much better result cannot be
expected. Indeed, we will show that for n > 3
ellipsoid E2n{a\, 3a\,..., 3a\) does not symplectically embed into the ball
the
B2n(A)
if A
3ay.
<
2, Lalonde and McDuff observed in [21] that Theorem 2
special case n
proved by their technique of symplectic folding. A refinement of their
In the
be
can
=
method will prove Theorem 2 in all dimensions.
Theorem 2
the sake of
a\
jr.
=
can
be
substantially improved by multiple symplectic folding.
clarity we restrict
optimal function
The
is the function
fEB
/£ß(a)
=
ourselves to dimension 4.
for the
J A \ E(n, a) symplectically embeds into B4(A)
fEß(a)
for
a
implies
that
/eb(ö)
The estimate
help
[it, 2tv]. For
the volume condition Vol
y/Wä.
embedding problem E(tv, a)
<^->-
inf
results with the
still
assume
\jt, 00[ defined by
our
a
can
on
We illustrate
=
We
>
2tt.
<
that
B4(A)
\.
1. In view of Theorem 1
have
of
Figure
>
2tt, the second Ekeland-Hofer capacity
This information is
(E(n, a))
/gß(fl)
a
For
<
a/2 + n
Vol
vacuous
(54(/es(<3)))
if
a
>
translates to
stated in Theorem 2 is obtained
we
47r, since
fEß(a)
>
by folding
Introduction
X
once.
We define the function seb
seb(o)
=
|A
inf
[it, oo[ by
| E(n, a) embeds into
It will turn out that
graph
on
(A) by multiple symplectic folding
B
sEß{a) is obtained from folding "infinitely many times". The
of the function seb is
computed by
a
computer program.
A
fEß(a)l
25tt
Figure
We
particularly
are
a —*
1: The result for the
oo.
embedding problem E(jz, a)
interested in the behaviour of
seb(2tt
+
the
Question
same
estimate holds for
1 How does
fEB(27v
+
2rc+ and
as
«Jïïci
Example 4,
cf.
3
2tt
<
6)
2n ? In
=
53 below. We in
Vol
lim
-
!
7
e
seb{o)
particular,
3
-2n
—
y/ïta
is bounded. This also follows from
Figure
-,
7
<
We shall also prove that the difference
—
—
near a
gô+
fEß(a)
-»
fsB-
/eb(o) look like
hm sup
e)
£
e->o+
so
as a
B4(A).
We shall prove that
limsup
and
fEß(a)
°->-
particular
Traynor's
Therefore,
theorem stated in
have
(E(7t, a))
«-ooVol(ß4(/£ß(fl)))
is bounded.
=
1.
(*)
Introduction
xi
In abuse of notation
of
(and
area a
open
not
we
denote in the
of radius
following by D(a) the open disc in R2
origin and by P(a\,..., an) the
centered at the
a)
symplectic polydisc
P{a\, ...,an)
P2n(a,...,
The "«-cube"
also be
to the
applied
D(a\)
=
x
x
D(an).
C2n(a). Symplectic folding
a) will be denoted by
can
variation of Problem 2.
following
Problem 3 What is the smallest cube
C2n(A)
into which U
symplectically
em¬
beds?
While
embedding
U into
a
minimal ball is related to
(a 1-dimensional, metric quantity), embedding U into
the
of its
minimizing
a
its diameter
minimal cube amounts
to the
projections
symplectic coordinate planes (a
see
symplectic quantity);
Appendix B for details. We will
therefore also study symplectic embeddings of ellipsoids and polydiscs into cubes.
We refer to the body of the thesis for the results.
to
minimizing
2-dimensional,
We
now
areas
more
discuss
dimensional ball
our answer
can
be
This also follows from
from
a
asymptotically symplectically
such
volume Vol
a
Example 4, which she obtained
Symplectic folding, however, can be used
connected symplectic manifold (M2n, co) of finite
-^ fMcon.
=
E
p„
FaK
identity (*), a four
by skinny ellipsoids.
method.
result for any
(M, co)
filled
theorem stated in
Traynor's
Lagrangian folding
to prove
to Problem 1. In view of the
(M, co)'
For
a
> it we
Vo\(kE(7T,...,7t,a))
sup
—
Vol
x
where the supremum is taken
define
over
(M, w)
all those k for which
kE2n(jz,
...
,7t,a)
sym¬
plectically embeds into (M, co). We define p„(M, co) in a similar way by replac¬
ing the ellipsoid E2n{it,..., it, a) by the polydisc P2n{it, ...,it,a).
Theorem 3
Assume that
(M, co) is
a
connected
symplectic manifold of finite
volume. Then
lim
Pq(M,co)
=
1
and
lim
p^(M,co)
=
1.
Introduction
Xll
We
finally answer a question about symplectic embeddings of balls
B2(tt) x R2"-2. Recall the
symplectic cylinder Z2n(jt)
standard
symplectic capacity
Definition An extrinsic
with each subset S of
axioms
Al.
into the
—
are
R2
a
number
c(S)
e
(R2,
on
[0, oo]
coq) is
in such
a
a
map
associating
following
c
way that the
satisfied.
Monotonicity: c(S)
such that
<p(S)
<
if there exists
c{T)
a
symplectomorphism
<p of
R2
C T.
A2.
Conformality: c(kS)
A3.
Nontriviality:
0
=
k2c(S)
c(B2(tt))
<
<
for all k
R
e
\ {0}.
oo.
it. An example of a nor¬
normalizing c we may assume c(B2(tt))
malized extrinsic symplectic capacity on R2 is the outer Lebesgue measure Jl on
R2. There exist normalized extrinsic symplectic capacities on R2 which are larger
than JL, however. E.g., for any subset S of R2 we define
After
z(S)
Then
z
=
—
is
inf {a
a
|
there is
a
symplectomorphism 
Hofer
=
of R
v)
e
R2
I
u2
+
v2
on
=
with
R2,
1
cp(S)
C
B2(a)}.
and for the circle
)
tt for the first EkelandJZiS1). Similarly, ci(S!)
for
the
tt
e(Sx)
displacement energy e, see [17].
=
and
capacity c\, see [6],
Appendix B.2 for more information on symplectic capacities on R2.
By the Nonsqueezing Theorem stated in Example 1, there does not exist a
symplectic embedding of the ball B2n(a) into the cylinder Z2n(n) if a > tt. So
fix a e ]0, tt], and fix a normalized extrinsic symplectic capacity c on R2. As
we shall see in Corollary B.10 of the appendix, the Nonsqueezing Theorem is
equivalent to the identity
=
We refer to
infc(/>(^(ß2n(a))))
where q> varies
Z2h{tt),
consider
over
all
symplectomorphisms
of
R2"
which embed
B2n(a)
B2(tt)
Z2n{n)
projection. Following [24]
sections of the image (p(B2n(a)) instead of its projection, and define
and where p:
Ua)
=
inf sup
—>
is the
c
[p (cp(B2n(a)) DxJj
n
into
we
Introduction
where cp
into
Xlll
again
Z2h{tt)
varies
over
and where
all
Dx
R2"-2. Clearly £c(tf)
symplectomorphisms
Z2u(tt)
C
R2"
of
which embed
denotes the disc Dx
=
B2n(a)
B2(tt)
x
{x},
that for
Moreover, Lemma 1.2 in
[23] implies
tt the
symplectic capacity c on R2 for which c(Sl)
the
On
her
search
for
is
Theorem
to
tt.
identity Çc(tt)
equivalent
Nonsqueezing
symplectic rigidity phenomena beyond the Nonsqueezing Theorem, D. McDuff
x
G
<
a.
any normalized extrinsic
=
=
therefore asked for lower bounds of the functions
It
as a -> 7T.
was
known to L. Polterovich that
and whether
Cc(ß)
Çc(a)/a
—>
0
as a ->
t,c(a)
0,
see
—»•
tt
again
[24]. We shall prove
Theorem 4
£c(a)
=
0
for
all
a
G
]0,
tt
[.
organized as follows: In Chapter 1 we prove Theorem 1 and sev¬
eral other rigidity results for ellipsoids. In Chapter 2 we prove Theorem 2 by sym¬
plectic folding. In Chapter 3 we use multiple symplectic folding to obtain rather
satisfactory results for embeddings of 4-dimensional ellipsoids and polydiscs into
4-dimensional balls and cubes. In Chapter 4 we look at higher dimensions. We
will concentrate on embedding skinny ellipsoids into balls and skinny polydiscs
into cubes. The results in this chapter form half of the proof of Theorem 3, which
is completed in Chapter 5. In Chapter 5 we shall also notice that for certain sym¬
plectic manifolds our embedding methods can be used to improve Theorem 3. In
Chapter 6 we recall the Lagrangian folding method invented by Traynor, and com¬
pare the results yielded by symplectic and Lagrangian folding. In Chapter 7 we
prove Theorem 4 and its generalizations.
Appendix A contains a proof of Proposition 1. In Appendix B we clarify the
The thesis is
relations between the invariants defined
by
Problem 2 and Problem 3 and other
review the Extension after Restric¬
Appendix
symplectic
tion Principle and prove an extension of this principle to unbounded domains.
Appendix D provides computer programs necessary to compute the optimal em¬
beddings of 4-dimensional ellipsoids into a 4-ball and a 4-cube obtainable by our
invariants.
C
In
we
methods.
We write
Lebesgue
\x\ for the Euclidean
measure
all manifolds and
symplectic
of
an
norm
open set (/ C
diffeomorphisms
forms and maps.
are
of
a
point
x
e
R" and \U\ for the
R". We work in the
assumed to be
C°°-category, i.e.,
C°°-smooth, and
so are
all
1
Rigidity
Denote
by 0(n)
dow each U
Recall that
&
G
we
R2" diffeomorphic
the set of bounded domains in
(n)
write
with the standard
symplectic
\U\ for the volume
^rô-
structure coo
=
to a ball. En¬
J2"=i
d*i
A
dyt.
of U with respect to the Euclidean vol¬
£)(n) be the group of symplectomorphisms of R2" and
Dc{n) respectively Sp(n; R) the subgroups of compactly supported respectively
ume
form fio
linear
=
Let
symplectomorphisms
R2w.
Define the
following
relations
U 
There exists
a
•
There exists
a
cp
e
£>(n) with <p(U)
<=>
There exists
a
symplectic embedding
U <3 V
Comparison
1.1
of
Clearly,
with
0(n):
on
C V.
<p(U)
C V.

V.
of the relations <,
 <2 =^ <3- It is, however, well known that all the relations <,
are
different.
Proposition
Proof.
1.1 The relations <t
are
That the relations <i and <2
(1.3) below and Example
but
following simple
are
different.
different follows from the
equivalence
4 of the introduction.
The construction of sets U and V
the
all
observation.
G
&(n) with U <3 V but U & V relies
Suppose
that U and V not
in
only
on
fulfill U <$ V
|V|. Thus, if U <2 V
symplectomorphic, whence,
particular, \U\
and cp is a map realizing [/ <2 V, no point of C" \ U can be mapped to V, and
3 V. In particular, the characteristic foliations on dU
we conclude that <p(dU)
and dV are isomorphic, and if dU is of contact type, then so is 3V (see [18] for
basic notions in Hamiltonian dynamics). Let now U
B2n(jr), let
are
=
=
=
SD
be the slit disc and set V
=
=
D(tt) \ {(x, y) |
B2n (tt) D (SD
x
•
•
x
•
>
x
0, y
=
0}
SD). Traynor proved in [34] that
are not even diffeo¬
symplectomorphic to B2h(tt).
morphic. For n > 2 very different examples were found in [8] and [5]. Theorem
1.1 in [8] and its proof show that there exist U, V G ö(n) with smooth convex
forn
<
2, V is
But 3[/ and 3 V
2
1.
boundaries such that U and V
are
symplectomorphic
but the characteristic foliation of 3 U contains
of 3 V does not. And
one
U
e
&(n),
n
Corollary
A in
plectomorphic
particular
and
We in
C°-close
that
see
set V G
for U
even
and C°°-close to B
(tt),
isolated closed orbit while the
an
[5] and its proof imply that given any
2, with smooth boundary dU of
>
Rigidity
contact
whose
&(n)
being
type, there exists
boundary
ball, the relation <3 does
a
a
sym¬
is not of contact type.
not
imply
the relation <2.
1.2
Rigidity
ellipsoids
1.1 shows that in order to detect
Proposition
must pass to a
ellipsoids
for
small
subcategory
described in the
8(n)
=
some
of sets: Let 8(n) be the collection of
=
E(a\,
an)},
...,
a
=
and write ^ for the restrictions of the relations <,• to
4l
the Extension after
(see [6]
=^3
are
or
the
=^2 =»
=^3
E(8a) =42 E(a')
=>•
C for
details).
It
(a\,
folding construction
general. But let us first
...,
an),
8(n). Notice again that
(1.1)
Since
ellipsoids
for all 8
is, however,
G
]0,
an
prove
general
a
common
[
starlike,
(1.2)
additional condition,
of Section 2.2 suggests that =^2 and =^3
and
1
are
known whether =^2 and
not
While Theorem 1.4 proves this under
the
symplectic
•
actually very similar:
Restriction Principle implies
Appendix
same:
=>
are
E(a) 43 E(a)
we
introduction,
{E(a)
The relations =^2 and =^3
via the relations <,
rigidity
are
different in
rigidity property
of these
relations:
Proposition
Proof. The
titivity,
1.2 The relations 4i
relations
clearly
partial orderings
reflexive and
on
transitive,
8(n).
so we are
left with iden-
i.e.
(E(a)
Of course, the
one
are
are
of =^1. It
4i
E(a')
identitivity
and
E(a')
of 4.3
implies
might, however,
4i
E(a))
the
one
be instructive to
=»
E(a)
=
E(a').
of =^2 which, in turn,
give
direct
proofs.
implies
the
1.2.
Rigidity
for
ellipsoids
3
It is well known from linear
symplectic algebra [18,
E(a) 4i E(a')
in
<^>
at
a[
<
p.
40] that
for all i,
(1.3)
=<li is identitive.
particular
Given U
G
0(n) with smooth boundary dU, the spectrum a(U) of U is
defined to be the collection of the actions of closed characteristics
clearly
invariant under
£>(«),
and for
cr(E(au...,an))
an
ellipsoid
it is
on
dU.
It is
given by
{d1(E)<d2(E)<...}
=
{ka{ |
:=
k
N, 1
e
<
i
<n}.
(1.4)
realizing E(a) 42 E(a'). The relations E(a) 42 E(a') 42
\E(a')\, and we conclude as in the proof of
E(a) imply in particular \E(a)\
dE(a'). This implies a(E(a))
a(E(a')) and
Proposition 1.1 that cp(dE(a))
the identitivity of =^2 follows.
To prove the identitivity of 43 we use Ekeland-Hofer capacities [7]. For the
Let
now
cp be
a
map
=
=
readers convenience
recall the
we
Definition 1.3 An extrinsic
ating
with each subset S of
following
axioms
are
=
symplectic capacity
R2n a number c(S)
Monotonicity: c(S)
A2.
Conformality: c(kS)
A3.
Nontriviality:
0
<
<
=
k2c(S)
for alU
c(B2h(tt))
capacities
symplectic capacities on R2n.
given by its spectrum (1.4):
and
form
For
(1.2)
we
by conjugating
the
given
conclude that for any 8\, 82
R
D(n)
map
a
c
associ¬
way that the
such that
=
c(Z2"(7r))
=>
/
>
1, of extrinsic
these invariants
{1, 2, 3}
and k
>
(1.5)
(1.6)
by À-1. Recalling
]0, 1[ the postulated relations
E(a) 43 E(a') 43 E(a)
are
0
E(ka) 4i E(ka').
map <p with the dilatation
g
C T.
< 00.
{dl(E)<d2(E) <...},
e
cp(S)
\ {0}.
countable
[7, Proposition 4]. Observe that for any i
E(a) 4i E(a')
g
g
family {c;},
symplectic ellipsoid E
a
a
{ci(E)<c2(E)<...}
seen
a
[0, 00] in such
if there exists cp
c(T)
The Ekeland-Hofer
This is
G
is
satisfied.
Al.
see
(R2", coo)
on
4
1.
Rigidity
imply
E(8281a) 42 E(8\a') 42 E(a).
Now the
monotonicity property (Al) of the capacities and the set of relations in
a'. This completes the proof of Proposition 1.2.
(1.5) immediately imply that a
=
D
The
equivalence (1.3), Example 4
ple show that =^2 does not imply =^1
and the Extension after Restriction Princi¬
in
general. However,
a
suitable
pinching
condition guarantees that "linear" and "non linear" coincide:
Theorem 1.4 Let
k
]§,&[
g
Then the following statements
(i)
B2n(K)
4i E(a) 4i
E(a')
(ii)
B2n(K)
42 E(a) 42
E(a') 42 B2n(b),
(Hi)
B2n(K)
43 E(a) 43
E(a')
We should mention that for
are
equivalent:
B2n(b),
4i
43
B2n(b).
proved in [10]. That proof
uses a deep result by McDuff, stating that the space of symplectic embeddings
of a ball into a larger ball is unknotted, and then uses the isotopy invariance of
symplectic homology. However, Ekeland-Hofer capacities provide an easy proof
The crucial observation is that capacities have
in contrast to
as we shall see.
symplectic homology the monotonicity property.
n
2, Theorem 1.4
=
was
-
-
In view of (1.1) it is enough to show the implication
Proof of Theorem 1.4:
(iii) =$ (i). We start with showing the implication (ii) => (i). By assumption,
B2h(k) 42 E(a) 42 B2n(b). Hence, by the monotonicity of the first Ekeland-
Hofer
capacity
c\ we obtain
k
and
by
the
monotonicity
(1.7) and
=
k
>
ci(E(a'))
whence
cn(E(a)),
—
a't
(1

b, whence the only elements in
<
<y(E(a)) possibly smaller than b
that an
a]
of cn
k
The estimates
<
are
a\,..., an.
ci(E(a))
=
ai
(1
(1.8)
It follows therefore from

for
polydiscs?
follows
now
of 43. Indeed,
5
by
implication (1.2)
shows that for any
in view of
We
83
in the
proof of the identitivity
>
G
B2n(b),
]0, 1[
E(8xa')
b/2
B2n(b).
42
we can
apply
the
already proved
B2n(b),
4i E(828ia) 4i E(8xa) 4i
be chosen
arbitrarily close to 1, the
completes the proof of Theorem 1.4.
can
( 1.3). This
use
as
43
8\, 82, 83
that <53<52<$i/c
large
so
E(a')
42 E(828ia) 42
B2n(838281K)
and since 8\, 82,
reasoning
43 E(a) 43
B2n(83S2S1K)
Choosing 8\, 82, 83
implication to see
similar
from
starting
B2n(K)
the
a
statement
(i) follows
D
Theorem 1.4 in order to prove Theorem 1 of the introduction, which
in the notation of this section reads
Theorem 1.5 Assume that
an
>
<
an
anda„
Therefore,
lence
<
an) 43
B2n(A) for
we assume
some
E(a\,... ,an) 43
2a\. A volume comparison shows
B2n(a\)
(1.3) imply
contradiction
43 E(a\,
that an
<
A
<
an.
Then
implies for n > 3 that
cally embed into B2n(A)
C3
...
,an) 43
B2n(A),
by observing
if A
<
for
some
]^, A[.
Theorem 1.4 and the
equiva¬
<
A. Hence,
>
2a\,
a1
as
claimed.
that the third Ekeland-Hofer
ellipsoid E2n(a\,
the
B2n(A)
G
a1
A. This contradiction shows an
We conclude this section
1.3
...,
2a\.
Proof. Arguing by
A
E(a\,
3a\,... ,3a\) does
not
capacity
symplecti¬
3a\.
Rigidity for polydiscs?
The
rigidity results for symplectic embeddings of ellipsoids into ellipsoids found
in the previous section were proved with the help of Ekeland-Hofer capacities.
Recall that P(a\,
an) denotes the open symplectic polydisc. We may again
<
<
<
assume a\
a2
an. The Ekeland-Hofer capacities of a polydisc are
given by
...,
Cj(P(au...,an))
=
jau
j
=
1,2,...,
(1.9)
6
[7, Proposition 5], and
analogues
of the
harder to prove.
i,
the
a
linear
>
r
enough
morphism given by
to
prove the lemma for
.
e
\z't\2
The
right
(Z\,Z2)
l->
nr2)
we
P(x,
<
^(kil2
hand side of
P(a\,
and
only
Then there exists A
embeds into the cube
...
It is
of the
polydisc
therefore either wrong
symplectomorphism if
1 + V2.
,tt, irr2)
polydisc P2h(tt,
linear
a
symplectomorphism.
(z\, Z2)
are
Many
a\.
Rigidity
,an)
...
if at
<
or
much
embeds
Ax for all
shows.
following example
Lemma 1.6 Assume
For
ellipsoids
area
It is for instance not true anymore that
P(A\,..., An) by
Proof.
the smallest
see
results for
rigidity
into
as
they only
so
1.
=
=
such that the
P2n(A,...,
2. Consider the linear
A) by
symplecto¬
1
.
(ZX,Z2)
|z2l2
—j=(Z\+Z2,Z\-Z2)-
=
have for i
+
n
C2n(A)
Ttr2
<
+
=
1, 2
2|zi||z2|)
\ 'j
+
<
+
(1-10)
r-
(1.10) is strictly smaller than r2 provided that
r
>
1 +
\[2.
D
Moreover, it is
discs instead of
not known whether the full
ellipsoids
holds true. Let
P(n)
analogue
of
tP(n) be the
Proposition 1.2 for poly¬
collection of polydiscs
{P(au...,an)}
=
and write <t for the restrictions of the relations <, to P (n), i
<2 and :<3
are
very
similar, again
all the relations <,
are
=
clearly
1, 2, 3. Again
reflexive and
transitive, and again the identitivity of <2, which again implies the
one
of ^1,
equality of the spectra, which is implied by the equality of the
an)
(Observe that, even though the boundary of a polydisc P(a\,
smooth, its spectrum is still well defined and given by er (P(a\,..., an))
follows from the
volumes.
is not
{ka{ I
...,
—
k
G
IN,
1
<
i
<
n}.)
For
n
=
2 the
identitivity
of ^3 follows from the
monotonicity of any symplectic capacity, which show that the smaller discs are
equal, and from the equality of the volumes, which then shows that also the larger
discs are equal. For n > 3, however, we don't know whether the relation ^3 is
identitive. In particular, we have no answer to the following question.
1.3.
Rigidity
for
polydiscs?
Question 1.7 Assume
P(a\,a2,a3)
°->-
Is it then true that «2
7
that there exist
P(a\,
=
a'2, a'3)
#2 an<^
a3
=
We also don't know whether the
symplectic embeddings
and
P(a\,
a'2, a'3)
c-^-
P(a\,a2,a3).
a3-?
polydisc-analogue of Theorem 1 or of The¬
orem 1.4 holds true. The symplectic embedding results proved in the subsequent
chapters will suggest, however, that the polydisc-analogue of Theorem 1 holds
true, see Conjecture 6.10.
1.
Rigidity
2
Proof of Theorem 2
2.1
Reformulation of Theorem 2
Recall from the introduction that the
ellipsoid E(a\,
...
,an) is defined by
n
E(a\, ...,an)
i2
i
E^
a
clearly
can
E2n(n,
Then
e) for every e
be reformulated
follows:
a) symplectically embeds
,tt,
...
as
0.
>
The
symplectic folding construction of Lalonde and Mc Duff considers a 4-ellipsoid
fibration of discs of varying size over a disc and applies the flexibility of vol¬
ume preserving maps to both the base and the fibers. It is therefore purely four
as a
dimensional in nature. We will refine the method in such
prove Theorem 2.1 for every
n
>
2.2 Assume
>
a
way that it allows to
2.
We shall conclude Theorem 2.1 from the
Proposition
a
2tt. Given
e
following proposition in
>
0 there exists
a
dimension 4.
symplectic
embed¬
ding
E(a, tt)
O:
«^
B4
(-
+
tt
+
e)
satisfying
7t\^(zi,z2)\2
We recall that
show that
•
^
77"
+
g
2i
+
2
Zl
|2
+tt\z2\2
2.2
implies
2.3 Assume that <E> is
permutation E2n(TT,
x id2n-4 to E2n(a, tt,
forall (zuz2) eE(a,TT).
a
| denotes the Euclidean
Proposition
Corollary
the
O
|
<
...
...
Postponing
norm.
the
proof,
we
first
Theorem 2.1.
as
in
Proposition
E2n(a, tt,
,tt, a)
,tt) embeds E2n(TT,
—>
..
2.2.
Then the
composition of
with the restriction
,tt)
.,tt,o) into B2n
...
(|
+
tt
+
of
e).
10
2. Proof of Theorem 2
Let
Proof.
definition
z
(zi,...,zn)
=
of the
(2.1)
E2n(a,TT,... ,tt). By Proposition 2.2
G
and the
ellipsoid,
n
TT
|$
id2n-4(z)\2
X
=
[\®(Zl,Z2)\2 + J2\Zi
TT
2
i=3
\Zl\~
TT
a
v^
,
,
i=2
as
claimed.
It remains to
proof Proposition
preparations.
The flexibility
of 2-dimensional
struction of the map O. We
prescribing
D(a)
it
on an
now
(i)
(ii)
point
p
g
a
make
preserving
sure
are
neighbourhood
equal
spectively.
to loops.
family
Xu
is
a
Then there is
corners
can
family
may describe such
of embedded
loops.
a
con¬
by
map
Recall that
origin.
loops in a simply connected domain
diffeomorphism ß: D(\U\) \ {0} —>
mapped
of the
a
to elements of
origin ß
is
a
U C
U
R2
\ {p}
is
for
X,
translation.
simply
symplectomorphism
connected domains in
of loops
R2 of
in U and V,
between U and V
re¬
mapping loops
(ii) imposed on the families taken into con¬
be weakened. Some condition, however, is necessary. Indeed, if
regularity
condition
of concentric circles and
and width
mapping loops
start with some
maps is crucial for the
and let X\j and Xy be admissible families
Remark 2.6 The
sideration
we
centered at the
Lemma 2.5 Let U and V be bounded and
area
that
we
U such that
concentric circles
in
a
area
area a
X of
family
called admissible if there is
In order to do so,
and nested
exhausting
denotes the open disc of
Definition 2.4 A
some
2.2.
to
larger
loops
than
a
positive
Xy is
a
family
constant, then
is continuous at the
origin.
of
rectangles with
no bijection from
smooth
U to V
O
2.1. Reformulation of Theorem 2
Proof of Lemma
11
2.5. Denote the concentric circle of radius
by C(r). We may
assume that X\j
{C(r)}, 0 < r < R. Let ß be the diffeomorphism parame¬
terizing (V \ {p}, Xy). After reparametrizing the r-variable by a diffeomorphism
of ]0, R[ which is the identity near 0 we may assume that ß maps the loop C(r)
of radius r to the loop L(r) in Xy which encloses the domain V(r) of area nr2.
We denote the Jacobian of ß at rel(p by ß'(rel(p). Since ß is a translation near the
origin and U is connected, det ß'(rel<p) > 0. By our choice of ß,
r
=
\V(r)\
=
=
detjS'
/
/
=
JD(nr2)
Differentiating
in
r we
pdp
det
p2n
/
=
det
Jo
ß'(rei(p) dcp.
Define the smooth function h: ]0, R[ xR
—»
R
as
(2.2)
the
unique
solution of the
problem
fth(r,t)
l/detj8'(re,'A(r,°).
=
Ä(r, 0=0,
depending
ß'(pei(p) dcp.
JO
JO
obtain
2tt
initial value
p2n
pr
p
jxr2
on
the parameter
R
t
0
=
(2.3)
We claim that
r.
h(r,
* e
t +
2Tr)
=
h(r, t)
+ 2tt.
(2.4)
strictly increasing in the variable t, that
for every r fixed the map h(r, •): R —> R induces a diffeomorphism of the circle
R/27TZ. In order to prove the claim (2.4) we denote by to(r) > 0 the unique
lit. Substituting <p
solution of h(r, to(r))
h(r, t) into formula (2.2) we
1, that
obtain, using detß'(reih(rJ)) j-(h(r, t)
It then follows, since the function h is
=
=
=
•
rtoir)
2tt
dt
=
=
Jo
Hence
h(r, 2tt)
h(r, t), solve
The desired
=
the
27r.
same
Therefore, the
diffeomorphism
a:
two functions in t,
initial value
U\{0}
is
-»
to(r).
problem (2.3),
now defined by
V\{p},
reiv)
h»
ß(reih(r>v))
h>
obtain for the determinant of the Jacobian
we
^(r,cp)-r. det ß'(reih^)
-
where
we
again
have used
bourhood of the
origin
proof of Lemma
2.5.
Consider
The set
a
in
C2
F(U,f)
is the trivial fibration
defined
f(z\)
1. The
a
translation in
a
a
punctured neigh¬
origin.
This finishes the
continuous function
f
:
U
-+
R>o-
by
eU,
tt\z2\2
<
f(zi)}
having as fiber over z\ the disc of capacity f(z\).
!F(U, f) and !F(V, g), a symplectic embedding <p : U "-+
U
over
symplectic embedding
g(cp(zi)) fox &\\ zi G U.
Examples
is
extends to the
smoothly
a
<
a
{(zi,z2) e<C2\zi
=
Given two such fibrations
defines
(2.3). Finally,
and thus
bounded domain (/cC and
F(U, f)
1,
=
dcp
r
x
cp
id: !F(U, f)
°->-
!F(V, g)
if and
only
if
2.7
ellipsoid E(a,b)
E(a, b)
2. Define the open
R(a)
can
F
=
be
represented
as
(D(a), f(zi) b(\- ^~
=
trapezoid T(a, b) by T(a, b)
=
{z\
—
(u, v) | 0
< u
<
=
Jr(R(a), g), where
a, 0 <
v
<
1}
b(\
T(a, a). The
g(u)
u/a). We set T4(a)
rectangle and g(z\)
useful
think
It
will
is
be
of
to
T(a, b) as
example
inspired by [22, p. 54].
very
O
depicted in Figure 2.
is
a
—
—
In order to reformulate
later
on
—
Proposition 2.2
allows to work with
more
—
we
shall prove the
convenient
"shapes".
following
lemma which
V
2.1. Reformulation of Theorem 2
fiber
13
capacity
b-
Figure
Lemma 2.8 Assume
e
>
2: The
0. Then
(i) E(a, b) symplectically embeds
T4(a) symplectically embeds
(ii)
Set e'
Proof.
struct
an area
=
trapezoid T(a,b).
ae2/(ab
+
ae
+
into
be).
+ e, b +
T(a
into
B4(a
We
+
are
e),
e).
going
preserving diffeomorphism a : D(a)
image R(a),
—>•
to use Lemma
2.5 to
con¬
R(a) such that for the first
coordinate in the
u(a(z\))
see
Figures
an
+
e'
for all z\
e
(2.5)
D(a),
3 and 4.
Figure
In
Tt\zi\2
<
"optimal
world"
Constructing
3:
we
would choose the
the boundaries of the
R(a)
as
If the
family
X
=
{Lu}
the
rectangles
a map â,
induced
with
we
embedding
loops Lu,
corners
0
a.
< u
<
a,
in the
image
(0, 0), (0, 1), (u, 0), (u, 1).
would then have
u
(â(zi))
< tt
\zi
\2
14
2. Proof of Theorem 2
for all
(zi, zi)
missible
(2.5).
R(a). The
e
family
X in such
admissible
e'/8
centered at
(uq, vq)
edges parallel
gles
a rectangle by specifying its lower
considered have
boundary
the
of the
with
We define
R(a).
e'
e'\
^- +
from below
(a
s
—
smoothing
boundary
of
For
right
<
r
e'/S
ad¬
the
as
loops
all rectan¬
following,
1
jA,
—
Lq be the
and let L i be
loops Ls by linearly interpolating
e'
.
a, the
with
rectangle
area
corners
e'
+
{U"l-4a<
Let
corner.
(^f-,
and
of the
(
Since us
4-aS)>
enclosed
'
by Ls
[°' 1]'
is estimated
by
e'\
{Ls},s
family
a
.
^-1.
us
Let
( j, ^ )
and
(l-s)^(l-S)4-a)
=
left and upper
corners
between Lq and L\, i.e., Ls is the
where us
to an
of Lemma 2.5
proof
In the
(uq, vq).
perturbed
satisfies the estimate
a
in the
(|-, ^).
=
be
can
to the coordinate axes. We may thus describe
rectangle
of
boundary
X
appearing
therefore the circles centered at
are
family
way that the induced map
a
Indeed, choose the translation disc
the disc of radius
L(r)
non
(
4J\
e'\
1
—
3e'
2—
>
us
4aJ
(2.6)
.
4
[0, 1 [, be the smooth family of smooth loops obtained from
g
the
corners as
indicated in
Figure
3.
By choosing
the smooth
[Ls}
by
corners
rectangular as s —> 1, we can arrange that the set U0<^<1 Ls
is the domain bounded by Lq and L\. Moreover, by choosing all smooth corners
rectangular enough, we can arrange that the area enclosed by Ls and Ls is less than
e'. Complete
e'/4. In view of (2.6), the area enclosed by Ls is then at least us
the families {L(r)} and {Ls} to an admissible family X of loops in R(a) and let
a: D(a) —>- R(a) be the map defined by X. Fix (z\, zi) 6 D(a). If a(zi) lies
on a loop in X \ {Ls}0<s<i, then u (a(zi))
<
\ < tt \zi\2 + e', and so the
of
Ls
more
and
more
—
required
area
(2.5)
a:
by
enclosed
is
again
D(a)
a
estimate
->
picture
(2.5)
by Ls
into
T(a
\z\\2,
and
like in
Figure
e,b
+
so tt
G
\z\\2
Ls for
+ e'
>
some s
In the
sequel,
we
>
us
the construction of
a
G
u
]0, 1[, then the
(a(zi)), whence
symplectomorphism
will illustrate
a
map like
a
4.
proof of (i)
we
E(a,b),
e). Take (z\, Z2)
g
a(z\)
completes
R(a) satisfying (2.5).
(z\, zi)
+
tt
satisfied. This
To continue the
for every
is
is satisfied. If
so
shall show that
that the
G
(a(z\),Z2)
symplectic
map
a
E(a,b). Then, using
s
T(a
+ e, b +
e)
id embeds
E(a,b)
the definition
(2.1) of
x
2.1. Reformulation of Theorem 2
15
E(a, b), the estimate (2.5) and the definition of e'
\
;
a
we
a
\
a
(
+
a
)
e
V
+
)
a + e
J
;
\
tu
=
xA
e) I 1
,
(fc +
e
)+e
e
a
M1
-
a
u(a(zi))\
öl
=
find
a
+
a
+
/
e
,
(a
v
+
/\
e);
—u(a(zi))
+e
e
"(«(zi))
;
a + e
\
It follows that
(ot(z\),zi) eT(a
as
e,b
+
e)
F
=
(R(a+ e), (b + e) (I
a
+
e
claimed.
In order to prove
from
to
+
a
D(a
(ii)
we
rectangular neighbourhood
+
Such
a
of
preserving diffeomorphism
an area
R(a) having
smooth
corners
and
area a
+
co
e
e) such that
tt\co(z\)\2
world"
shall construct
map
we
co can
again
<
u
+
e
forall z\
be obtained with the
would choose the
(u, v)
=
help
R(a).
of Lemma 2.5. In
Lu in the domain
loops
admissible family to
G
R(a)
as
(2.7)
an
"optimal
before. This time,
we
an admissible family X of loops as illustrated
perturb this non
in Figure 4. If the smooth corners of all those loops in X which enclose an area
greater than e/2 lie outside R(a) and if the upper, left and lower edges of all these
loops are close enough, then the induced map co will satisfy (2.7).
Restricting co to R(a) we obtain a symplectic embedding co x id: T4(a) c—>R4. For (zi, zi) e T4(a) we have tt \z2\2 < a(\
(u, v) e
u/a), where zi
=
—
R(a).
In view of
(2.7)
tt
we
conclude that
(|û>(zi)|2 lz2|2)
+
<
—
=
u
+
e+a{\--^j
u
+
e
a
+ e,
+
a
—
u
16
2. Proof of Theorem 2
D(a).
a
D(a + e)
co
Figure
and
so
(co
x
id)(zi, zi)
4: The first and the last base deformation.
e
B4(a
Lemma 2.8 allows to reformulate
Proposition
2.9 Assume
a
e) for all (zi, zi)
+
Proposition
2tt. Given
>
e
2.2
as
T4(a).
D
follows.
>
0, there exists
ey
(zi,z2)
a
symplectic embed¬
ding
V:
Z\
—
T(a,n)
(u, v) and
u'+
tt\z'2\2
z\
<
—
Corollary
a
T4(^
we
+
h>
(z\,z'2),
l-7r|z2|2
Proposition
2.9
implies Proposition
of Proposition 2.9 holds true. Then
E(a, tt) ^-^ B4 (| + tt + e) satisfying
statement
^
2+ +
forall(u,v,zi)&T(a,TT).
(2.8)
a
symplectic embedding
<
TT
first show that
2.10 Assume the
tt\<P(zi,z2)\
+
(u', v'), satisfying
-+ e-\
2
Postponing the proof,
exists
<-+
<î>
2i
:
2.2.
there
i2
kin
+
7r|z2|2
forall (zi,z2)eE(a,TT).
(2.9)
2.1. Reformulation of Theorem 2
Let e'
Proof.
proof
0 be
>
of Lemma 2.8
x
a
satisfying
id:
so
17
small that
ca
+ e'
we can construct a
E(ca, ctt)
T(ca
°->-
2tt, where
>
c
1
=
e'/tt.
—
As in the
symplectic embedding
+
e',
e')
+
ctt
T(ca
=
e', tt)
+
the estimate
u(a(z\))
<
tt\z\\2
-—t,
+
can
and another
for all zi
+ ae' + ne1
D(ca)
e
(2.10)
symplectic embedding
co x
id
T4
:
(—
+
tt
+
e')
(—
B4
<-»
+
2e'\
+
tt
satisfying
tt\co(zi)\2
Since
ca
replacing
+ e'
e
>
<
u
^
for all z\
(u,v)
=
2tt, Proposition 2.9 applied
guarantees
*:
(zi,Z2)
+ e'
a
T(ca + e',n)
(zi, zi)
tt
(—
+
+ e'
ca
+
tt
e')
replacing
(2.11)
•
a
e'/2
and
+
tt
+
e')
,
(*i(zi, zi), *2(zi, Z2», satisfying
(u(a(z\)),
v,
Cfl
,
—+e'+
<
embeds
7Tm(0!(zi))
2
,
ca +
T(ca + e', tt). Set
zi)
symplectically
e
(y
T4
^
o
Then $
to
R
symplectic embedding
M(vI/1(Q;(Zl),z2))+7r|vl/2(a(z1),z2)|2
for all
G
into
E(ca, ctt)

=
54
,
(2.12)
e'
(co
x
id)
+
7r
+
(y
o
+n \z2\2
o
ty
2e;).
o (a x id).
Moreover, if
£(ca, C7r), then
|0(zi,z2)|2
=
TT\co(^x(a(zi),Z2))\2 + Jr\^2(a(zi),Z2)\2
(2.11)
<
(2.12)
ca
<
TT
+ 2eH
2
(2.10)
,
u(Vi(a(zi), zi)) +
ca
<
2
+n
nu(a(z\))
;
ca +
—
e'
|2
ca
+ e'
ca
7T2|zi |2
2
ca
a(e')2
tt
-^--\
,
\-2e'-\
9
Z2)\z
+TT\Z2\
tt2\zi\2
n
|*2(a(zi),
,
—
ca
+ e'can + ae'+
,
tt
e'
i2
h7r|z2|
18
2. Proof of Theorem 2
where in the last step
7r+3e
that
< tt
E(a,Tr)
O:
+
G
U
again
we
+
O
ca
10(Z1,
(Vc)_
=
E±^-)
E(a,Tr) and by
TT
used
+ e'
Z2)|2
O
o
B4
by */c
^/c.
o
(f
+
the choice of
e',
C
2tt. Now choose e'
>
We denote the dilatation
R4 by
->•
E(a, tt) into B4
(z\, zi)
e.
+
tt
e),
c
\ 2
t
+
—
2
and define
tt
|zi
|2
<
embeds
a
T
2i
7T
+
É
,9
,
-,
+
!
a
Zl
i2
!
,9
,
h 7Tk2l
a
E(a,TT). This proves the required estimate (2.9), and
(zi, Z2)
Corollary
It remains to prove
2.10 is
the
D
2.9. This is done in the
following
two sections.
construction
The idea in the construction of
the small fibers from the
other. As in the
so
complete.
Proposition
folding
for all
—+7r|z2|2
c
2
The
and since
l7r2|zi|2
+
c
a
<
2.2
by *Jc,
small
so
a
3e'
a
=
proof
0
>
symplectically
TT2\Z\\
.
„
of
Then $
also
-|$(VCZ1,>/CZ2)|
=
1 /ca
for all
R4
in
large
embedding
ones
section
previous
^
as
in
Proposition
2.9 is to separate
and then to fold the two parts
we
on
top of each
denote the coordinates in the base and the
fiber
by
z\
Step
1.
Following [22, Lemma 2.1] we first separate the "low" regions over R(a)
"high" ones. We may do this using Lemma 2.5. We prefer, however, to
=
(u, v) and
an
Z2
=
(x, y), respectively.
from the
give
an
explicit
Let 8
Figure
>
construction.
0 be small.
Set ¥
!F(U, f), where U and /
=
5, and write
f
a
+
p2
=
un\u>
L
=
U\(P1UP2).
tt
+ 115
are
described in
2.2. The
folding
construction
U
_z_
P%
Pi
2±2L
§+5
5:
Figure
+ 115
a
the low fibers from the
Separating
+
\
large
+ \28
fibers.
Hence, U is the disjoint union
u
Choose
a
smooth function h
(i) h(w)
(ii) h'(w)
(iii) Ä
(|
(iv) h(w)
1
=
52)
=
w
0 for
<
+
for
=
h(a
[0,
a
+
8]
->•
—
as
in
Figure 6,
i.e.
[0, f ],
G
g
+
<52[,
w
G
]§, f
w) for all
we
[0,
a
+ 8].
have that
f+52
dw
<
8
and
h(w)
We may thus further
(v) h(w) <8 for
]0, 1]
5,
By (ii), (iii) and (iv)
/,
to
:
p1\Jl]Jp2.
=
require
w G
]§
/.
J%+s-8i
that
+
82, f
+ 5
-
ä2+S
<52[,
ö?u»
h(w)
<
8.
20
2. Proof of Theorem 2
h(w)
w
Figure
6: The function h.
Consider the map
ß: R(a)
R2,
1
(u, v)
H>-
h(w)
io
Clearly, ß
is
a
symplectic embedding.
ß\{u<l]=id
and
We
ß]{u>^s}
see
=
dw, h(u)v
from (i), (iv) and (vi) that
id+(^ U8,0).
(2.14)
+
(2.13) and (v) imply that ß embeds R(a) into
These identities and the estimates
region in R(a) is mapped to the black region in U,
and so on). Finally, by construction, ß xid symplectically embeds T(a,n) into !F.
U
(cf. Figure
7, where the black
2
% + 8
2
Figure
Step
a
2.
a
a
7: The
embedding ß
We next map the fibers into
similar way
morphism
a
as
it
was
used in the
mapping D(tt)
to the
a
:
R(a)
convenient
a±n
°->-
+m a+7t + n8
U.
shape. Using
proof of Lemma 2.8 we
rectangle Re and D (|) to
Lemma 2.5 in
find
the
a
symplecto¬
rectangle
with
2.2. The
smooth
folding
corners
construction
/?,-
as
21
in
specified
7T|z2|2
+ 25
Figure
8. We
require
that for Z2
£
D
(|-)
y(a(zi))-(-7^-28),
>
i.e.
y(o(z2))
7r|z2l
<
Figure
Write for the
resulting
bundle
(idxo)!F
In order to fold
S(Pi)
=
over
then turn it in the z\ -direction
Step
c :
R
3. In order to
—
[0,
1
—
move
28] be
|
I(t)
=
fQ
x
S
for Z2
-
smooth
0,
(lt\
\—j
(2.15)
.
corners
S(PX)\\*(L)\\S(P2).
=
we
first
move
S(Pi) along the y-axis and
S (Pi).
we
cut
off function
t <
f
l
D
the fibers.
$(P2) along the y-axis
a
e
a)!F of rectangles with smooth
$(P\)
over
C{t)~\\-28,
Set
-
Preparing
8:
(id
n
2
+ 28
+ 38<t
and
again [21,
Figure 9:
follow
as
in
t > s±2-
<^-
c(s) ds and define the diffeomorphism
+
cp
+
p.
105,
98,
:
R4
—>
R4 by
355]. Let
22
2. Proof of Theorem 2
c(t)
1 -28
-
f+
^±ZL
25
Figure
</?(«,
x,
v,y)
(
=
9: The cut off
+
v
w, x,
(
c(u)
x
+
t
+ 105
c.
-
J
,
y +
I(u)
(2.16)
We then find for the derivative
d<p(u,
whence cp is
defining
a
x, v,
the number
0
A
I2
with A
1^ by I^
id
=
that 5
and
-^
we
I
(^^
c(u)
c(u)
0
of the criterion in
+
105),
^{u^+ios}
we
[18, p. 5]. Moreover,
find
=id +
(0,0,0,Iœ),
(2.17)
compute with the help of Figure 9 that
TT
<
=
*
TT-+25<
assuming
h
symplectomorphism in view
V\{u<i+2*}
and
y)
TT
-+28<
loo
<
,
2
The first
inequality
in
-
+ 58.
(2.18)
(2.18) implies
cp(P2
x
R{)
fl
(R2
x
Re)
0.
Remark 2.11 The map cp is the crucial map of the
(2.19)
folding
deed, cp is the only map in the construction which does
not
of 2-dimensional maps. It is the map which sends the lines
the characteristics of the
(the
cut
split
{v,
x, y
as
a
In¬
product
constant}
to
Re
O
hypersurface
(u,x,y)
which generates
construction.
off
h^
of)
\u,x,c(u)\x
the obvious flow
1
+ -),y
separating /?,•
from
2.2. The
folding
construction
23
4. From the definition
(2.16) of the map cp and Figure 5 and Figure 8 we read
off that the projection of <p(S) onto the (u, u)-plane is contained in the union XL
Step
of U with the open set bounded
by
two lines
=
5 +
u
<
c(u)
[u
<
—
1
—
(a + tt)/2
map y is the
a/2
+
5} and {u
5. Define
+
115, 0
identity,
isometry
left
of Pi. In
edge
<
and
which maps the
local
a
v
on
<
the
(a
of
graph
+
tt)/2
+
u
1
U D
}
follows: On Pi
as
{u
>
of
right edge
particular, we have
=
a/2
P2
for zi
a
+
+
=
-
{«
(u, v)
as
shown in
10:
black square <S
Figure
u'(y(u, v))
-
{u
<
a/2
+
0
<
8} the
>
(a
G
P2,
+
tt)/2
+
115}
to the
(2.20)
U\
Figure
looks
U n
=
{(u, v) \
+ \28-u.
Uq
remaining
10. Observe that
y of Ii into
25} it is the orientation preserving
U D
=
c(u), the «-axis and the
115}, cf. Figure
symplectic embedding
u'(y(z\))
On the
5 +
h*
Folding
=
V. Pi
in the base.
{a/2
10. We then have for
(| 5)
+
<
I
+ 105
-
+ 8
< u
(u, v)
e
a/2
+
\ (Pi
U
<
U
25} the map
Pi),
(u (| + 5)) + 5,
-
y
24
2. Proof of Theorem 2
i.e.
TT
u'(y(u,v))
the map y
By (2.19)
10
as
82/3.
Figure
6
we
1 + 39
choose
(2.21)
cp(S).
preserving embedding
follows: Set uq
=
Similar to
on
and orientation
an area
be
Figure
proved
215/2. Moreover, set /
tt/2+
can
1-a+ 135.
—u-\
id is one-to-one
x
The existence of
<
=
a/2
+ 25 and u\
5/4 and choose A.3
a
>
0
smooth function h:
so
[|
as
proposed
(a
=
+
n)/2
small that
+
5,|
+
k3l
in
+
<
25]—>-
]0, 1] such that
for
§
(i) h(u)
=
1
(ii)/z(M)
=
^forMG[|
a
(iü)
The
Jl
,31
>
1
j+S
1
h(w)
embedding
u near
+ 5 and
+
dw
f,| + f
f2
and
:
S
->•
[|
2
+ 25,
f],
+
-rj-^dw
,„,
.,
2"1"
ys
|
u near
"•"
+ 5, u0 +1 +
=
I
s
f]
x
[0, 5] defined by
a
(u,v)
M*
I
-
+ 5 +
I
Ja+S
and illustrated in
Figure
'ML
ö
11 is
dw, h(u)v
symplectic.
i
y^
^-3^
f
+ 5
Figure
MO + ^ "0 + / +
«o
§
11: The map yg.
domain S' in the
(z/, i/)-plane
painted in
Figure 10: By the choice of / we may require that the part of the "outer" boundary
of £' between (uq, 0) and (u\, 1), which contains (u\, 0), is smooth, has length
/, and is parametrized by Ç(s), where the parameter s e I := [uq, «o + ^] is arc
length and
We
now
map the
h(w)
image
Ç(s)
=
(s,0)
Ç(s)
=
(u\+uo
of ys to
+ l
-
a
s,\)
on
[uo,u\],
on
\uo
+ l
-
\,uo + l\.
as
(2.22)
2.2. The
folding
construction
Denote the inward
k\
>
0
pointing
so
G
s
differential
for
some
second
/ is
a
>
equation
-*
%(s,t)
=
f(s,0)
=
R2,
(s,t)
Then
0-
in
f(s, t)
that the
(s, t)
dety^CM)
area
R2
preserving.
for
identity
and
[0, A2].
t G
We
now
+
preserving,
tv(s)
we
consider the initial
(2.23)
s near
—
at
smooth solution
g
I
x
half of
a
tubular
(s, t) detdr](s, f(s,t))
choose the parameter A3
(2.22)
A.2], and yf is
[0,
>
/
x [0, A2]
[0, A.2]. This and the
on
(u', v')
i->
In view of the identities
uq and t
G
/
composition
(s, f(s, t))
\->
df
=
a
k\ for all (s, t)
<
(2.23) imply
:
yields
diffeomorphism of / x [0, k2] onto
Moreover, by the first equation in (2.23),
in
area
Ç(s)
^
0
a
i.e., y^ is
We choose
v.
\/detdr](s,f(s,t))
with parameters
yK
is
along £ by
parameter. The existence and uniqueness theorem for ordinary
equations
A2
[0, Aj]
x
In order to make the map
embedding.
value problem
an
in which
unit normal vector field
small that
rj: I
is
25
an
neighbourhood
=
of Ç.
1,
for £, the map yç is the
isometry
for
s near
«0
+1
0 in the construction of ys smaller than k2.
Restrict yf to the gray
region 7x]0, A3[ in the image of ys, and let y^ be the
smooth extension of y? to the image of ys which is the identity on {u < uq} and
an isometry on {u > uq + I}. By (i), the composition
yf o ys is the identity near
u
a/2 + 8 and an isometry near u
a/2 + 28. It thus smoothly fits with the
defined
the
at
beginning of this step.
map y\u\B already
—
—
finally adjust the fibers. In view of the constructions in Step 2 and
Step
projection of the image <p(S) onto the Z2-plane is contained in a tower
shaped domain T (cf. Figure 12), and by the second inequality in (2.18) we have
T c {(x, y) I y <
once more our Lemma 2.5 we construct a
j + 45}. Using
symplectomorphism r from a neighbourhood of T to a disc such that the preimages of the concentric circles in the image are as in Figure 12. We require that for
Step
5. We
3, the
26
2. Proof of Theorem 2
Figure
Z2
=
1 to
map ^
<
y +
TT\r(z2)r
<
.-l
nW~
—>
Step
T(a,n)
:
^
tt\t(z2)\2
D(tt)
a :
Step
2.3
the tower to
Mapping
disc.
a
(x,y),
where
vj/
12:
fory>-|
+ 38
+
-r
c^-
are
R4
the
ingredients
is defined
and set 5
=
min{^, f^}.
as
of
the
Proposition
our
that for all (u,
u'
(2.25)
Re,
construction. The
folding
composition
=
folding
of maps
(y
x
t) ocpo(ß xcr).
2.9. So let
e
>
0 be
We define the desired map *I>
that *I> meets the
verify
(u, v,x,y) G T(a, tt)
to show
e
(2.26)
proof
Recall that it remains to prove
mains to
for z2
+ 85
Re is the diffeomorphism constructed in Step 2.
5
End of the
(2.24)
TT
(Z2)\
(idxx)o(y xid)ocpo(idxo)o(ß xid)
=
25,
and write
v,zi)
+
e
estimate
as
as
in
in
So let
(2.8).
required
(«', v'', z2)- By the
^(z)
=
Proposition
z
(2.26).
=
It
(zi, zi)
2.9
re¬
=
choice of 5 it suffices
T(a,rc)
7T\z'2\2-rr\z2\2
<
-
+ 155.
(2.27)
2.3. End of the
proof
We
three
U
distinguish
Pi
=
ß(zi)
according to the locus of the image ß(z\) in the set
TJ P2 (see Figure 5 and Figure 7). We denote the M-coordinate of
]J L
Case 1.
=
=
=
Indeed, the definition of the map ß illustrated in Figure 7 shows that if
u"(ß(u, v))
u
Pi. The first identity in (2.17) implies <p\s(Pi)
id, and Step 4
id. Therefore, u'
u"(ß(u, v)). Moreover, u"(ß(u, v)) <
ß(zi)
implies y\s(Pi)
u
cases
ß(u,v)byu"(ß(u,v)).
=
+ 8.
27
<
f,
then
|. Summarizing,
>
u"(ß(u, v))
we
=
—
7T|Z2|2
id
we
map
r
|t (<T (Z2)) |2
TT
=
<
find
<
u
TTU
,
,9
+TT\z'2\
a
e
]§, § +5],
then
+ 8.
Re and z2
<r(z2)
=
ï(o~(zi))- Hence, the
yields
TT
\a~l (a (z2)) f
Finally, we have u < | + 5. Indeed, if «
(2.14) implies ß(u, v) G P2. Altogether we
u'
u"(ß(u, v))
and if
have
u'
Using again (p\s{px)
estimate (2.25) for the
u,
-Tt\Z2\
>
+
|
|
+ 5, then the second
can
TT
=
\z2\2
|
+
+ 85.
identity
in
estimate
/
9
+ 85
TT
TT\
)+5 + -+85
<
u{\
<
TT
TT\
/
-(1--) +
2 V
2
a)
V
a)
2
a
-
+105
a
Case 2.
ß(zi)
g
P2. By the second identity in (2.17)
(2.20), u'
have
(p\s(p2)
u'(y(ß(z\)))
we
=
id +
identity
f+
125
u"(ß(u, v)). Moreover, u"(ß(u, v)) > u + | + 105. Indeed, the definition
of ß shows that if u"(ß(u, v)) > ^L + 125, then u"(ß(u, v))
u +
f + 115,
and if u"(ß(u, v)) e \^f- + 115, sÇL + 125[, then u < f + 5. Summarizing, we
(0, 0, 0, /a,),
and so, in view of the
=
=
a
+
—
=
have
u'
Step
2 shows
a(zi)
s
Ri, and
so
<
a
—
u
y(<r(z2)
+ 28.
+
(0,
/oo))
>
-f
-
25.
Hence, the
28
2. Proof of Theorem 2
estimates (2.24), (2.15) and (2.18)
tt\z'2\2
=
7r|r((r(z2)
<
y(cr(z2))
have
u
|.
>
TTU
,
u'
+
Indeed, if
we can
,9
\-tt\z2\
a
+ 38
I00 +
+
+
+
+ 35
TT
|2
7r|z2r
Pi. Altogether
g
(0,/oo))|2
+
=
we
+
7^
(*lz2|2-f) (! 55) !
<
Finally,
ß(u, v)
imply
+ 85.
-
u
|,
<
then the first
identity
in
(2.14) implies
estimate
-tt\z2\
/
9
a-u[\ +
<
V
a
/
TT
TT\
—
)+28-\
a/
TT\
TT
a--(l + -) +
<
2 V
h 85
2
-
2
a)
+ 105
a
=
Case 3.
ß(z\)
G
Using
L.
u'
Since
7r|z2|2
<
(0,1(u"(ß(u,
estimate
7r|z2|2
lit)
=
we
v)))))
<
(1
-
<
25)(*
o-fe)
-
-
+
(tt|Z2|2 I)
-
tt
|z2|2
have
+
+
(1
-
>
|
+
I(u"(ß(u,v))))\2
+
25)
+
^
+ 38
(u"(ß(u, v))
-
ü-
-
28) + | + 35
u"(ß(u, v))---28- 28u"(ß(u, v))
u"(ß(u, v))
y{cr(zi)
(2.24) and (2.15) and the
25)) read off from Figure 9 yield
I(u"(ß(u,v)))
<
+
\-a + 135.
25. Hence, the estimates
(f
(2.21) implies
Rt, cf. Figure 8. In particular,
e
TT\r(x(a(z2)),y(cr(z2))
y(cr(Z2))
we
-u"(ß(u, v))-\
have
-f
>
+ m
the definition of cp, the estimate
<
=
Finally,
f,
5
+ 5
by
the definition of
+ 8a +
L, and
u
452
>
+ 35.
| by
the
2.3. End of the
proof
first
(2.14). Altogether
identity
in
TTU
,
+
u'
a
,9
tt\z2\
29
-n\z2\
we can
-,
estimate
TT
„
<
TT a
-u"(ß(u,v)) + -+a + \38
2
+u"(ß(u, v))
-
°-
a
-
25
-
25
2
(| 5)
+
+5a + 48l + 35
a
=
-
2
+ 145 + 25z
a
<
2
where in the last step
we
proof of Proposition
Recall that
in view of
252
have used that
We have verified that the estimate
and the
l5S'
+
2.9 is
(2.27) holds for all (u,
the
proof
v,
^.
<
zi)
T(a,n),
complete.
by Corollary 2.10, Proposition
Corollary 2.3,
5 which follows from 5
<
2.9
implies Proposition 2.2,
of Theorem 2.1 is
and so,
complete.
A-,
Figure
13:
Folding
an
ellipsoid
into
a
ball.
Remarks 2.12
1. As the verifications done in this section
maps
the
ß,
a, cp, y and
required
estimate
x
constructed in the
(2.8).
showed, the specific choice of the
previous
section is crucial for
obtaining
30
2. Proof of Theorem 2
2. We recall that the
struction is the
<$>
=
=
where
embedding
composition
c_1
o
(&>
x
id)
o
Î>
c~
o
(co
x
id)
o
(id
c
is the dilatation
o
by
(a
x
a
x
r)
O
/a1)
o
(y
:
E(a,sr)
°^-
54(|
+
tt
+
e) in
our con¬
x
id)
(a
o c
x
id)
o
cp
o
(id
x
a)
o
(ß
o
x
id)
number close to 1.
folding map *I> : T(ü,tt) <^-> T4(A) can be visualized as in Figure 13.
Essentially, * restricts to the identity on the black rectangle and maps the triangle
{u > |} to the light triangle and the triangle {tt\z2\2 > §} to the dark triangle.
3. The
o
c,
3
Multiple symplectic folding
In four dimensions
shall
we
exploit
the great
only depend on the fiber coordinates
results for simple shapes.
We first discuss
in four dimensions
symplectic
rather
provide
to
of
flexibility
maps which
satisfactory embedding
modification of the
a
symplectic folding construction
explain multiple folding, and finally calculate
ellipsoids and polydiscs into balls and cubes which
de¬
scribed in Section 2.2, then
of
optimal embeddings
by these
be obtained
the
can
methods.
In order not to disturb the
(arising
sequel. Since
the
exposition unnecessarily with the arbitrarily small
"rounding off corners" and so on) we shall skip them in
from
5-terms
all the sets under consideration will be bounded and all
structions will involve
only finitely
many
steps,
con¬
will not lose control of the
we
5-terms.
The map
tated
a
in
2 of the
Step
the estimate
by
(2.8)
one
Figure
of
S(P\), and
we
13. In order to
is not necessary,
we
Step
this
modify
of
now
of
er
y
as
in
Step
^
image
can
the
room
disjoin
in Section 2.2
result,
Z2-projection
for
n
=
composite
in
2, in which
case
n
of
define cp
a-1
on
the
r on a
of
image
map *!> defined
in
as
T4(a/2 + tt
+
image
neighbourhood
a.
If all the 5's
(2.26)
e)
Figure 15. Essentially, ^
triangle with vertices
for
2. As
a
Ä(P2) from
triangle
Replace
as
dic¬
was
>
the estimate
follows:
cp will almost coincide with the
of *l> will be contained in
as
the
folding construction as
given by Figure 14. If we
the floor Fi and maps the white
floor F2.
3 had to
4 and define the final map
be visualized
given
the
such that it restricts to
appropriately,
the
construction
ended up with the unused white sandwiched
use
Step 2 by the map a
Z2-projection of the image of
a
folding
construction
used to obtain the 2n-dimensional
consequence, the map <p of
the
folding
Modification of the
3.1
in
(2.8)
the map
in
(2.16),
of
a.
Choose
of the
were
the
image
chosen
will be one-to-one, and
some
small
restricts to the
(|, 0),
(a, 0),
e.
The map
identity
(|, |-)
on
to the
32
3.
Figure
Multiple symplectic folding
14: The modified map
in four dimensions
a.
A-.
f
Figure
3.2
3tt
Folding
a
in four dimensions.
Multiple folding
Neither Theorem 2
the
15:
A
Traynor's theorem stated in Example 2.2 tells us whether
ellipsoid E(tt, 4tt) symplectically embeds into the ball B4(A) for some A <
(cf. Figure 53). Multiple symplectic folding, which is explained in this sec¬
nor
symplectic embedding of E(tt,4tt) into 54(2.6927t). To
understand the general construction it is enough to look at a 3-fold. Up to the
final fiber adjusting map x, the folding map ^ is then the composition of maps
explained in Figure 16, in which the pictures are to be understood in the same
sense as the picture in Figure 2: The horizontal direction refers to the base and
tion, will provide
a
the vertical direction indicates the fiber.
the
ellipsoid E(a, tt)
as
the
Here
are
the details:
trapezoid T(a,n) fibering
over
the
Fix
a
>
tt
and
rectangle R(a)
=
3.2.
Multiple folding
33
ß
x
id
id
x er
o satisfying
u\,..., u\ G
embedding E(a, tt)
heights U,
into
cube. Then define the
i
tt
=
a
i
Yli=i
ui
=
a>
ball and in 3.4.1 for
=
1, 2, 3, by
1,2,3.
i=i
Step
1
(Separating
smaller fibers from
larger ones).
Let U and
/ be
as
in
Figure
17.
Proceeding as in Step 1 of the folding construction in Section 2.2 we find a sym¬
plectic embedding ß : R(a) <-^ U such that (ß x id)(T(a, tt)) C ^(U, f).
Step 2 (Preparing the fibers). The map a is explained in Figure 18. More pre¬
cisely, a maps the central black disc to the black disc D, and up to some neglected
5-terms
we
have
h—h
y(a(zi))
In
general,
h
-
h-h +
=
h
-
IZ2I2
n\z2\2
TT |Z2|2
* |Z2|2
tt
for
most
Z2 g D
(I3) \ D,
for most Z2
e
D
(l2) \
D
(l3),
for most Z2
e
D
(h) \
D
(l2),
for most Z2
G
D
(tt) \ D (h).
folding n times, a maps the circles in the n + 1 annuli around a
disc alternately to rectangular loops with essentially constant maxi-
when
small central
+
34
3.
l\
Multiple symplectic folding
Figure
mal but
decreasing minimal
constant
Step
3
minimal but
(Lifting
abbreviate
<p:
(ß
x
c(t)
the
=
increasing
Y^,i=i c*'(0
°->-
cp(u,x, v,y)
Step
4
(Folding). Step
1. The
17:
u3
h
R4
and
Ht)
is defined
4 in Section 2.2
rectangular loops
y-coordinate.
off functions q
=
as
u, x, v +
u
-
u
F(U, f).
maximal
cut
-
M4
y -coordinate and to
fibers). Choose
a)(T(a, tt))
l2
U2
in four dimensions
Jo
c(s)ds.
in
(2.16) by
c(u) [x
now
+
The
-
requires
over
)
with
Li, i
=
essentially
1, 2, 3, and
symplectic embedding
,y +
I(u)
three steps.
folding map y\ is essentially the map y of Section 2.2: On the part of
the base denoted by P\ it is the identity, for u\ u\ + l\ it is an isometry. Observe that by construction,
the slope of the stairs S2 is 1, while the slope of the upper edge of the floor Fi is
7r/a < 1. The sets S2 and Pi are thus disjoint.
3.2.
Multiple folding
35
Figure
2. The map
on
certain
y2xid
is not
a.
global product map, but restricts to a product
domain: It is the identity on Fi TJ Si TJ F2, and it is the
remaining domain, where y2 is explained in Figure 19: It
of its
pieces
18: The map
really
a
product y2 x id on the
is the identity on the gray part of its domain, maps the black square to the black
part of its image, and is an isometry on {u < 0}. The map y2 is constructed the
same
way
as
the map y in Section 2.2.
3. The map y3
It is
x
id, which
turns
isometry on F4, looks like
identity everywhere else.
an
to the
Step
5
(Adjusting
the
fibers).
F4
over
the map
F3, is analogous
given by Figure
10
to the map yi
on
x
id:
S3, and restricts
The
Z2-projection of the image of cp is a tower
shaped domain T. The final map x is a symplectomorphism from a small neigh¬
bourhood of y to a disc. It is enough to choose r in such a way that up to some
(*', y') e T
(x,y), z2
neglected 5-term we have for any z2
=
y<y'
This finishes the 3-fold
analogous. Here,
we
=>
folding
have u\,
=
|r(z2)|2
<
|r(z2)|2.
construction. The n-fold
...,
TT
/,
=
un+\,
a
^-^
}=\
construction is
heights
-r-~k
y
tt
folding
u,,
i
=
1,
...,«,
(3.1)
36
3.
19:
Figure
floors Fi,...,
in space
F„+i, and
specified
in the
stairs
Multiple symplectic folding
Folding
the left.
The thin stairs
S\,..., Sn.
following lemma,
on
in four dimensions
which is
require
proved in Steps
some
locus
4.1 and 4.2
above.
Folding
Lemma 3.1 Let
Si be the stairs connecting the floors Fi and Fi+i of
minimal respectively maximal height U, i
n.
1,
=
(i) If
i is
lower
odd, then the stairs Si
contained in
edge of length /,-, left edge of length 2/,-,
cf. Figure
(ii) If i is
are
...,
20
(i);
even, then
moreover,
no
the stairs Si
smaller
a
trapezoid with horizontal
right edge of length k,
and
trapezoid contains Si.
contained in
trapezoid with horizontal
upper edge of length U, left edge of length /,-, and right edge of length 2/(-,
cf. Figure 20 (ii); moreover, no smaller trapezoid contains Si.
are
a
3.3.
Embeddings
into balls
(i) Folding
on
the
right.
Figure
3.3
Embeddings
In this section
on
the left.
20: The locus of the stairs.
into balls
multiple symplectic folding to construct good embeddings
ellipsoids and polydiscs into four dimensional balls.
we use
of four dimensional
3.3.1
(ii) Folding
Embedding ellipsoids
into balls
A(a, u\)<
u\
u\ +
Figure
Recall from Theorem 1 that if
plectomorphic
to
a
E(a, tt), does
<
not
l\ A(a,u\)
21: A 12-fold.
2tt then the
ellipsoid E(n, a), which
symplectically embed into B4(A) if
is sym¬
A
<
a.
38
3.
We therefore fix
2tt. We
>
a
Multiple symplectic folding
again
think of
E(a, tt)
as
in four dimensions
T(a, tt) and of
B4(A)
T4(A)
T(A, A).
trapezoid T4(A) into which
T(a,Tr) embeds via multiple symplectic folding we have to choose the Ui's, which
appeared in Section 3.2, optimally. Our strategy to do so is as follows. We shall
as
In order to find the smallest
=
describe
a
procedure
which associates with each u\
/
A(a, mi)
and either
empty
or
a
2tt\
2tt + I 1
J
(3.2)
mi,
finite sequence u2, «3,..., ww+i and the attribute
admissible,
or an
finite sequence u2, U3,..., u^ and the attribute non-admissible. In both
the number N
cases
=
]0, a[ the number
g
=
N(u\) will depend only
u\, and the
on
procedure
will
symplectic embedding of T(a, tt) into T4(A(a, u\)) by (iV-fold) sym¬
plectic folding if and only if u\ is admissible. In view of a > 2tt and equation
describe
a
(3.2)
have to look for the smallest admissible u\. As
we
admissible if u\
we
an/(a
<
+
tt), and
u\ is admissible if u\
u\
will show that if u\ is admissible, then
follows that there is
a
unique
and u\ is non-admissible if u\
uq
=
we
shall see, u\ is
a/2. Moreover,
>
u\
is admissible for any
uo(a) such that
non-
>
u\.
u\ is admissible if u\
>
It
uq
uq. Therefore,
procedure yields symplec¬
embedding of T(a, tt) into T4(A(a, uq) + e) for any e > 0. Finally, we will
explain why our procedure is optimal, i.e., multiple symplectic folding does not
yield an embedding of T(a, tt) into T4(A) if A < A(a, uq).
<
our
a
tic
We start with
minimal
height
describing
our
procedure.
of the first floor Fi is then
Fix u\
l\
=
G
tt
—
Figure 21, we define A (a, u\) by the condition that
the "upper right boundary" of T4(A(a, u\)), i.e.,
We
now
r\
=
a
if and
—
only
T(a,n) into
embeddings
contained in
are
image
if r\
is admissible.
are
S{
u\. The
u\+2l\
<
u\,
T4(A(a,
forced to fold
2, in such
T4(A(a,u\)).
a
way that «(- is maximal
Define the
u\,u2 then describe
u\)) obtained by folding
one
remaining length
length
u2. Then the
a
once.
height
we
r\
T4(A(a,
=
symplectic embedding
Since u\
>
fold at u\
—
u2
of F2atwi— u2isl2
=
>
1\
by
u\))
r\, and u\
of
a/2, these
constructed in Section 2.2. If r\
second time. Assume that
in
\u\.
T(a, tt) after folding at u\ is contained in
i.e., u\ > a/2. If r\ < u\, we set u2
better than the
a
second floor F2 has
>
suggested
27r\
2tt + I 1
of
Indeed, the data
are not
—
to choose U{, i
successively try
and the stairs
—
As
(tt/a)u\.
The
at u\.
the second floor F2 touches
(
A(a, u\)
]0, a[ and fold
>
u\, we
0, i.e., the
—
(tt/o)u2.
3.3.
If
into balls
Embeddings
h
h>u\
u\, then
>
>
u\
—
U2, and so the
Folding
Lemma 3.1
(ii)
not contained in
are
T4(A(a,
52 is contained in
this condition is
(n/a)u2
—
shows that the stairs S2
39
is therefore
u\))
equivalent to
the condition
{u > 0}. A necessary condition that
tt
/] < u\. In view of l\
(tt/o)ui
=
on
—
u\
an
>
«1
(3.3)
.
a
+
n
If this condition is not met, then u\ is non-admissible. If
Lemma 3.1
(ii) shows that for
the maximal such U2 is
Zj
u\, then the
<
U2 small
enough we have S2
characterized by the equation I2 + U2
{u
C
=
>
Folding
0}, and
u\, which
that
by (3.1)
translates into the formula
+
a
u2
not
contained in
S2
C
T4(A(a,
remainder
by
Then
by (3.4).
mi)).
r2- If ^2
<
that
is
we
I3
=
fold at
l2
«2, we set «3
2I3, and
in
so
T4(A(a,
the
U2
U2
case
a
—
u\
—
>
<
2l2,
U3
2/2
>
then u2
of
—
«3
<2l2
(i) shows that
Lemma 3.1
2l2, and if this condition is
characterized
by
(2n/a)u3
the
equation
2/3.
=
u2
—
«3
Since
=
r\
=
S3
2/2
—
new
Indeed, the data
T4(A(a,
u\))
third time. Assume
«3, and its
—
U3
the stairs
height at I2 + «3
<2l2
(2n/a)u3
=
—
S3
are not
is contained in
not
a
a
U2 of the
—
into
T(a, n)
U2, we are forced to fold
A necessary condition that
—
U2
r2, and u\ is admissible.
symplectic embedding
>
is
u\ is non-admissible. Assume that
contained
T4(A(a,
u\)) is
met, u\ is non-admissible.
2/2, the Folding Lemma 3.1 (i) shows that S3
>
—
If u2
Folding
u\)).
therefore u2
a
=
0}, but it is still possible that S2
>
length
n
—
I2 + U3, i.e., the third floor F3 has length
(n/a)u3.
—
in which
twice. If r2
by folding
a
{«
C
(3.4)
.
n
—
We denote the
u\, U2, «3 then describe
obtained
S2
T4(A(a, u\)),
an
u\
a
We define «2
n
=
>
T4(A(a,
C
If
u\)) whenever
2n, the maximal such «3 is
(2n/a)u3, i.e.,
a
«3
z—(K2-2/2).
=
a
We define M3
Assume
a
—
that
X^=i
u\ is admissible.
of
T(a, n) into
forced to fold
(3.5)
2n
by (3.5).
now
have been chosen
length
—
we
folded
already
i
times, where i is
maximal, and that u\ is still
uj °f
me
Indeed,
remainder
u\,u2,
T4(A(a, u\))
again.
As in the
by
...,
/
possibly
If r,
<
ut,
ul+\ then describe
obtained
case
r,.
=
by folding
even, that U2,
...,
ut
admissible. Denote the
a
we
set ul+\
=
rt, and
symplectic embedding
/ times.
If r(
>
w,,
we
are
2, the Folding Lemma 3.1 (i) shows that if
40
3.
21,, then
<
u,
u\ is
for which the stairs
Multiple symplectic folding
non-admissible, and that if
S,+i
contained in
are
T4(A(a,
21,, then the maximal
>
u,
in four dimensions
ul+\
u\)) is
a
ul+\
=
—
a
We define u,+\
remainder
Denote the
by (3.6).
r,+\. If r,+\
by
<
—
length
The
again.
S(+2 C {u
equation
>
l,+2
ul+2 +
We define ul+2
=
l„
u,+\ +
in which
which
by (3.1)
+
this way
or not.
embedding
struction after
if u\
—
ul+\ of the
new
r,+\, and u\ is admissible.
of
T(a, n) into
case
C
the
translates into the formula
(3.7)
"i+in
—
{u
by
n
=
S,+2
0}, but possibly S!+2
>
u\ in non-admissible. If
to decide for
try
we
already
> a/2.
following
each u\
The value u\ is admissible if and
of
T(a, n) into
T4(A(a,
and u\ is non-admissible if and
have
r,
S!+2
is not contained
T4(A(a, u\)),
C
we
before.
Proceeding
an
Then
by (3.7).
T4(A(a, u\)),
missible
=
=
and that the maximal such ul+2 is characterized
0},
a
go
set ul+2
Uj
symplectic embedding
a
a
on as
X^'i=i
by folding i + 1 times. If r,+\ > ul+\ +I,,we are forced
Folding Lemma 3.1 (ii) shows that for ul+2 small enough we
Ui+2
in
—
(3.6)
obtained
to fold
have
(u, -21,).
a
+/,,we
ul+\
Indeed, u\, u2,..., ul+2 then describe
T4(A(a, mi))
2n
finitely
only
many folds
or
u\)) after
if the
if the
lemma
implies
we
only
an/(a
<
if
finite
procedure
procedure
seen, u\ is non-admissible if u\
Also recall that
a
]0, a[ whether it is ad¬
G
procedure leads to
number N(ui) of folds,
our
leads to
an
embedding
does not terminate. As
+
n),
ob¬
we
and u\ is admissible
have to look for the smallest admissible u\. The
that there exists
a
unique
uq
=
uo(a) in the interval
an
1(a)
:=
'
a
such that u\ is admissible if u\
u[
and N(u\)
Lemma 3.2 Assume that u\,
then
u\
is
We abbreviate ./V
Proof.
procedure
lemma
for
u\
admissible,
>
=
G
<
are met whenever
they
(3.8)
2_
]0, a[ and that
u\.
<
uq.
u\ is
admissible,
Assume that the
embedding
<
u\
If
N(u\).
N(u\) and N'
our
tt
uq and u\ is non-admissible if u\
associated with u\ is described
by going through
+
procedure
=
A^(mj).
u\,..., u^+i- We shall prove the
by
checking
and
are met for u\.
that the
folding
conditions
3.3.
Embeddings
u\
If
and
S2
>
2. Then
T4(A(a, u\))
C
41
is admissible and N'
N
so
into balls
u'x
is
>
1,
=
u\
we are
an/(a
I2 <
equivalent
to
done.
Otherwise,
Equation (3.4)
«2-
u[
<
u\
a/2,
<
Observe that the next condition
n).
+
>
shows
u'2
u'2,
>
u2,
and so
I2. Therefore, 1'2 < h < «2 <
equation (3.1)
1'2
If
N'
is
admissible
and
done.
are
we
T4(A(a,
2,
Otherwise,
S'2
u'x
Wj)).
u2 < u'2 < r'2 < r2, and so N > 3. In view of (3.1) we then have u'2 > u2 >
2/2 > 21'2, and equation (3.5) shows u'3 > U3. lfu'x is admissible and N'
3, we
are done.
>
n
>
a
whence
+
i.e.,
(1
Otherwise,^
u'3+l'2,
—n/a)(u\ +u'2) 2u'v
and
shows
now
<
C
=
=
—
a
—
n
(1
>
n/a)(u\
—
ceed in this way
S{
C
T4(A(a,
ui)
+
2«3, i.e.,
+
using equations (3.6)
fori
u\))
even
is
and
=
<
(3.9)
B4 (seb(<z)
>
We next
e
explain why
our
is optimal in the sense that we cannot
B4 (seb(ü)) by multiple symplectic fold¬
procedure
E(n, a) into a ball smaller than
Indeed, observe that our procedure
lows:
To A
2n +
(1
each A
of
—
G
and then choose «,-, i
2n/a)u\,
inside
staying
choose u\,u2,
T4(A).
therefore to choose
some
The
obstruction
>
equivalently
u\
arising
way of
procedure
leads to
Lemma 3.3 We have
and the
Proof.
procedure
u\
—
N(u\)
embedding
—>
00 as
—
By
u\
fol¬
A
=
our
S£ß(a),
/,• and
>
>
r-
obstruction after N'
\
uq. The
procedure
is
let u\, U2,..., u^
<
circumvent the
u\
<
Ui.
Then,
ri, and so the
N folds.
value uq is non-admissible,
associated with uq does not terminate.
Arguing by contradiction,
is wrong.
with u\
an
improving
<
our
however, the proof of Lemma 3.2 shows that /•
modified
«i(a, A) defined by
=
procedure, and try to
folding ./V times by choosing
by
as
maximal with respect to the condition
...
only possible
after
be described
2, maximal. In other words, for
of the u\ smaller. So let A
be the sequence associated with A
embedding
can
]2n,a[ associate the number
successively
we
embeds into the ball
0.
embed
ing.
3. We pro¬
(\-JL\ u0(a).
2n +
ellipsoid F(7r, a) symplectically
e) for any
>
w,-. The lemma then follows.
Lemma 3.2 shows that the
+
N
]2n, 00[ by
on
sEB(a)
so
(3.7) and observing that the condition
equivalent to /,•
Define the function seb(o)
I2, and
«3 +
>
7-3
assume
that the first statement of the lemma
Lemma 3.2, there exists iV such that
uq small
enough.
uq is so small that
N(u\)
In the
=
N(u\)
—
N for all u\
>
uo
sequel we assume that u\ > uq and that
N. By the proof of Lemma 3.2, the functions
42
3.
Ui(u\),
i
I,
=
m°
i
2,..., N
=
N, and u^(u\)
...,
lim
=
Ui(u\),
Assume first that
Case 1. iV
T4(A(a,
Case 2.
into
N
u2,
=
decreasing as u\ \ uq, and the functions k(u\),
increasing as u\ \ uq and bounded. We set
are
are
if
and
=2,..., N,
during
lim
=
the first N folds there is
no
then the
Folding
Lemma 3.1
U(u\),
embedding
shows that
(i)
i
I, ...,N.
=
obstruction at uq.
T4(A(a,
u2, then T(a, n) embeds into
>
in four dimensions
mo)) by folding
T(a,n) embeds
uo)) by folding twice.
3 and N odd.
>
T4(A(a,
u°N
If
/^_j
+
u°N+l, then T(a,n) embeds
u°N + l°^_l
u°N+l, then the
>
mo)) by folding N times, and if
Lemma 3.1
Folding
i
1. If «o
=
once, and if mo
into
1
—
Multiple symplectic folding
(i) shows that T(a, n)
=
T4(A(a,
embeds into
uo)) by folding
N + 1 times.
Case 3.
N
folding
imply
times,
N
that
conditions in
mo
>
=
u°N+l, then T(a, n) embeds into T4(A(a, uo)) by
2n
u°N+l, then the Folding Lemma 3.1 (ii) and
T4(A(a,
mo is admissible with
uo)) by folding
N(uo)
=
N
N + 1 times.
N(uo)
or
=
N + 1. Since all
procedure are open conditions, it follows that
enough, then mi is admissible with N(u\)
our
Mi is small
>
a
T(a, n) embeds into
Therefore,
—
u°N
and if u°N
If
even.
=
N + 1. This contradicts the definition of mo. So
if mi
<
N
N(ui)
that there is
or
mo and
=
embedding
appearing at the latest at the iV'th fold. We shall conclude the
proof of Lemma 3.3 by showing that an embedding obstruction at mo appearing at
the latest at the A^'th fold implies an embedding obstruction at all mi near moassume
an
obstruction at mo
Case I. mo
<
Therefore, u2
If N
2,
=
If Af
>
2,
an/(a
+
0 and
=
n). Then
so
N >2 and
>
«2(mi) for
we
find
U2(u\)
<
2h(u\) for
T4(A(a,
Si(uo) çt
a
uo)) for
If i
N,
we
find
u^+$u$
If i
<
N,
we
find
m,(mi)
(mo)
N,
we
contradiction.
<
near
2
/°
=
mo.
mi near mo, a contradiction.
i.e., there is
mi near mo,
an
embedding
mo; this is another contradiction.
=
obstruction for mi
=
=
M3(mi)
Case II.
=
=
find
near
If i
l®
an/(a + n), i.e., h(uo)
I®
mo-
=
we
obstruction for mi
Case III. ui
mo
<
>
some even
u^(ui) for
2l{(u\)
for mi
/, i.e.,
mi
if
near
near
>
u®
i.e.,
some even
i.
contradiction.
mo, a
mo,
for
there is
an
embedding
mo; this is another contradiction.
Z(- (mo) for
some even
find in view of
a
>
/. Then
2n that
u®
=
mat+i(mi)
2
if.
>
u^(u\) for
mi
near
mo,
3.3.
into balls
Embeddings
If
i
=
N
If i
N
<
if + lf+\
1, then urN+1
—
contradicting
mi near mo,
+
43
1,
—
if+2
We conclude that
admissible, and
uf+2
=
This contradicts
have
oo as
—>
u^+$u$
u^-$u$ for
<
if
0 and
lf+l
=
T4(A(a,
C
lf+2,
=
u\)) for
whence
mi near mo.
\ mo- The value mo is therefore nonI, Case II and Case III above, the proce¬
u\
in Case
seen
=
S(+2(«l)
dure associated with mo cannot lead to
thus
find
so we
i.
uf+]
N(u\)
as we
and
lN_\,
<
>
find
we
"?•
>
N
obstruction. Lemma 3.3 is
embedding
an
proved.
D
The computer program in
which
lowing lemma,
procedure does
Appendix D.l computing mo (a) is based
implies that the value mo is the only value for
By
Lemma 3.3
and
u,(uq)
sequences
F,(uo). Seth(uo)
:—
our
Then the
which
our
associated with u\ leads
procedure
after finitely many folds.
<
obstruction
embedding
Proof.
the fol¬
not terminate.
Lemma 3.4 Assume that u\
to an
on
uq.
associated with mo generates the infinite
procedure
1,(uq),
i
1, 2,
=
Xî^("o)-
...,
of
Thenh(uo)
h(uo)
=
lengths
and
heights
of the floors
A(ü,uq). We claim that
<
(3.10)
A(a, uo).
image of T(a, n) in T4(A(a, uq)) obtained by folding first
infinitely many times touches the upper vertex of T4(A(a, uq)),
In other words, the
at mo and then
i.e., the sequence F((mo) of floors creeps into the upper
cf.
Figure
assume
3.1
(ii)
21. In order to prove the
h(uo)
shows that
there exists
hi(uo)
A(a, uo). Then
<
>
j
h,(uo)
/
>
j, and
contradiction proves the
now
:=
w
U2i(uo)
N such that
e
w/2 for
Assume
+
identity (3.10),
u>
for all
hi(uo)
<
so a
Jl^li
l,(u\)
1,(uq),
i
=
l,(u\),
:=
1,2,
^/,(mi)
i=\
0. The
Folding Lemma
>
ui(uo)
>
j. Then M2((mo)
Jl%i
<
>
oo,
w
—
oo.
This
Since mi is
non-
u2i(mo)
=
i
=
some
<
«o-
mi does not terminate and generates
The
1,2,
mi
proof
of Lemma3.2 shows that
Therefore,
OO
h(u\)
T4(A(a, mo)),
contradiction and
identity (3.10).
admissible, the procedure associated with
>
>
of
1, 2,.... Since h(uo)
=
for i
vu/2
that Lemma 3.4 is wrong for
the infinite sequence
z
by
argue
A(a, uo)—h(uo)
>
=
we
corner
00
>
^Zi(wo)
i=i
=
h(uo)
=
A(a,u0)
>
A(a,u\).
44
3.
The contradiction
h(u\)
Multiple symplectic folding
A(a,u\) shows that Lemma 3.4 holds
>
While the computer program in
each
e
>
0 the value of
sEß(a) up
Lemma 3.5 The function seb
on
Appendix
D. 1
true.
computes for each
to accuracy e, the
into the function
qualitative insight
in four dimensions
a
lemma
following
>
2n and
gives
some
seb(o)-
]2n, oo[ is strictly increasing and hence almost
everywhere differentiable. Moreover,
seb is
Lipschitz
continuous with
Lipschitz
constant at most 1.
Proof.
Assume 2n
(a, uo(a'))
and
<
a
a'. In view of the procedures associated with the
<
(a', uo(a')),
l$a, uo(a ))
=
the
Mo(a )
n
<
n
a'
a
and
a
—
uo(a!)
<
a!
uo(a') imply
—
(
seb(o)
Since
In + I 1
=
a'
pairs
inequalities
that
2;r\
I M0(a)
Mo(a)
Mo(a )
/i(a
,
Mo(a ))
Mo(a'). Therefore,
<
/
<
=
2n\
2n + I 1
I
M0(a )
=
sEß(a ).
arbitrary, we conclude that the function seb is strictly increas¬
ing, and so, as every increasing real function, almost everywhere differentiable.
Assume again 2n < a < a'. We set 5
a'
a and mi
Mo(a) + 5. Since
a
<
were
=
Mo(a)
<
=
—
a, we have
Mo(a)
Ml
-^
<
(3.11)
—.
a'
a
It follows that
n
l\(a
,
u$
=
n
a'
n
Mi
Mo(a)
n
<
=
/i(a, Mo(a)).
a
procedures associated with the pairs (a, uo(a)) and (a', u\), this
a
uo(a) imply that mi > Mo(a')- Using
u\
inequality and the equality a!
definition (3.9) and inequality (3.11), we may therefore estimate
In view of the
=
—
SEß(a')-sEß(a)
=
—
(l
7\u0(a')-
(
-
y
2n\
~
v)Ul
,
—
u\
—
v
l\
(
~
r,
V
2n\
~
(ul
mi
=
a'—
—
tJ
Mo(fl)
uo(a)\
I
uo(a) —2n\
\a'
<
J u0(a)
Mo(a)
a.
a
)
3.3.
into balls
Embeddings
Since
<
a
a'
tinuous with
45
arbitrary, we conclude
Lipschitz constant at most 1.
were
that the function seb is
investigate
Proposition
the behaviour of the function
Fix
a
unique
a
>
2n.
value u\t2
Then M12 +
SEß(a)
on
any open interval.
27T+.
as a —>
3.6 We have
hm sup
Proof.
con¬
D
We don't know whether the function seb is differentiable
We next
Lipschitz
By
the
sEB(2n
proof
2u2(u\2)
a.
=
<
-
.
of Lemma 3.2 and
such that
Mi,2(fl)
=
3
e)-2n
+
In view of
N(u\t2)
=
Lemma 3.3 there exists
by
2 and
equation (3.4)
"3(1*1,2)
find u\^2
we
=
«2(^1,2)2
=
q
a3~f^
Therefore,
a
A2(a)
and
so
Â2(27r)
=
A(a,
:=
|.
Since
u\
2)
2n +
=
seb(o)
<
(a
—
27t)
-^2(0) for all
+
n
3a +
a
>
,
n
2n, the proposition
follows.
proof describes the
optimal embedding of E(n, a) into a ball obtainable by folding exactly twice.
More generally, we can compute the function A^(a) describing the optimal em¬
bedding obtainable by folding exactly N times as follows. For each a > 2n
and each N
1,2,... there exists a unique value u\^
u\jy(a) such that
Remark 3.7 The function
A2(a) constructed
in the above
=
=
N and such that mah-i(mi,n)
N(u\^)
un(ui,n) + In-i(ui,n) if N is odd
and MAf+i(Mijv)
wjv(Ml,;v) if N is even. If N is odd, we replace In-i(u\n)
the
by
expression given in formula (3.1). Plugging the expression for un+\ (uyj^)
into the equation
=
=
=
ul,N +
and then
function
u2(u\j^)
+
+
•
alternately using equations (3.6)
of a. We finally find
un+\(u\n)
and
(3.7),
(
An(o)
—
A(a, wi,#)
=
27T + 11
we
=
a
compute u\,n
27t\
I
u\^(a).
as a
rational
46
3.
Multiple symplectic folding
in four dimensions
instance,
For
1
A\(a)
2n +
=
(a
2n)-,
—
A2(a)
2
+
a
2n +
=
(a
—
2n):
n
3a+
n
and
,
(
,
A3(a)
Lemma
By
3.2, u\^+i (a)
in the
n
2n
=
of Lemma
,
>
+
(a
-
u\^(d)
<
3.5,
(a
o
2n)
7r)(a
+
4(a^
+
27r)
—r-
+
an
+
.
nl)
for every N and every
a
2n, and arguing
>
that the function
proof
increasing for
u\^(a)
1,2,... ,is therefore a strictly decreasing family
every N. The family {A at}, N
of strictly increasing smooth rational functions on ]2n, oo[ converging to seb(u).
as
we see
is
=
In view of Dini's
One
might try
jÂN(2n)
all
a
Theorem, the convergence is uniform
|
<
for
2n and N
>
a2("i,2(a))
some
>
N.
in
given
Â^(2n)
However,
on
bounded sets.
Proposition
=
|
3.6
for all N
by showing
2.
>
Indeed, for
2we have
«2(«i,jv(fl))
>
the estimate
improve
to
2l2(u\N(a))
>
>
2l2(u\2(a))
2n
—«2("i,2(a)),
—
a
and
we
so
lima^2jr+ u2(u\^(a))
=
lima^.2jr+ "2(M1,2(#))-
In view of formula
(3.4),
conclude that
lim
lim
uitN(a)
a-±2n+
mi
i-+2n+
a(a).
i
Next, the identity
'
AN (a)
implies
=
2n +
2n
{
a
JMl
for all
N(a)
that
d
2n
-A^(a)
da
=
-^ru\
a1
A?(a)
2ttn\
+ I 1
\
d
1 —Mi
J da
a
defines
is
rational function
expression for u\^(a)
not a singularity of u\n(o), the
2n. Taking the limit a —> 27r+
=
d
—AN(2n)
da
as
claimed.
we
\
=
—-
a
for all
jv(a)
The formal
a
2n
>
a
rational function
-jû\^(a)
2n a^2n+
uhN(a)
=
R.
>
2n.
Since 2n
is bounded
near
therefore find
1
hm
on
a
—
d
lim
2n a^>-2n+
u\t2(a)
=
—A2(2n)
da
3
=
-,
7
O
3.3.
Embeddings
into balls
47
We do not know how to
seb
(a)
the
subsequent
as a ->•
E(n, a) into
a
P(tt, § +n)
Proposition
directly.
oo
As
a
ball.
3.8 We have
—
\pna~
<
2n
seb(o) is further discussed
for all
and
a
>
compared
2n
.
with the result
yielded by
in 6.2.1.1.
Embedding polydiscs
have
asymptotic behaviour of the function
prove the following proposition at the end of
by comparing the optimal multiple folding embedding of
with the optimal multiple folding embedding of the polydisc
ball
into
Lagrangian folding
3.3.2
We shall
the
section
seb(<*)
The function
analyze
into balls
in the
proof of Lemma 2.8, the disc D(a) is symplectomorphic to
the rectangle R(a). The polydisc P(n, a)
D(n) x D(a) is therefore symplec¬
the
fiber
to
Since
x
R(a)
D(n).
D(n) over each point (u,v) G R(a)
tomorphic
is the same, the optimal embedding into a ball obtainable by multiple symplectic
folding is easier to compute for a polydisc than for an ellipsoid. In contrast to our
optimal embeddings of an ellipsoid into a ball, which were obtained by folding
"more and more often", the optimal embedding of a polydisc into a ball obtainable
by multiple folding will turn out to be described by a picture as in Figure 22. Our
embedding result stated in Proposition 3.10 below is readily read off from such
a picture. We aim, however, to show that this embedding result is the best one
obtainable by multiple folding. We therefore proceed in a systematic way.
we
seen
=
A3-,
H
M]
Figure
22: The
A3
optimal embedding P(n, a)
-
U
a
°->-
B4(A)
for
a
=
107T.
48
3.
Multiple symplectic folding
B4(A)
in four dimensions
trapezoid T4(A). Fix a > n. Folding
R(a) x D(n) first at mi g ]0, a[ determines T4(A(a, u\)) by the condition that
the second floor F2 touches the "upper right boundary" of T4(A(a, u\)). Then
A (a, Mi)
mi + 2n. We then successively choose m(-, i > 2, maximal with
the
condition of staying inside T4(A(a,u\)).
to
respect
We
think of the ball
again
the
as
=
The
Folding
Lemma 3.1
if necessary, is mi
if and
if mi
only
(ii) shows that
condition for
a
n, and that then the stairs
>
The
2n.
condition
S2
folding
a
T4(A(a, u\))
contained in
are
mi for
second time,
only
folding a
Folding Lemma 3.1 (i) shows that folding a third time, if
necessary, is then possible whenever M2 > 27r, i.e., mi > 3n. For N >2, the only
condition on mi for folding an Af'th time, if necessary, is mi > Nn.
If our procedure leads to an embedding obstruction after N folds, then choos¬
ing m- < Ui leads to an embedding obstruction after N' < N folds. It is therefore
enough to compare embeddings obtained from our procedure.
therefore u\
>
of
R(a)
D(n) into
x
number of folds needed. If mi
and
N(u\)
<
<
procedure
associated with mi leads to
N(Ul).
is odd and the last floor
{u
=
u'x
does not touch {u
Fn(U\)+i
u\
suchthat
<
Set N
=
N(u\).
Observe that
(mi)
is continuous and
on
=
1 and u^+i
admissible
admissible
u\
<
at
mi would be
u\
u'N+l
un- We have u^+i
R(a)
x
we
can
u\.
=
=
>
continuous and
does not touch
3.
>
If N
u^ + n,
already
admissible with
that un+1 touches the
so
u'2
would
Assume next that ,/V is
and
N(u\)
Then there exists
FNûi )+i
an
touches
increasing
N(ui)
=
in mi, and
in u\.
Fyy+i
if N
u^ + n
<
such that
Mj such that
however, folding
«j
that
those admissible u1 's for which
are
decreasing
Assume first that N is odd and
an
0}.
=
even or
u\ and such that N (u\) is odd and
N, the functions U2(u\),..., u^(u\)
if N
N(u\) is
0}.
Proof.
«N+i
an
T4(A(a,u\)). We then write N(u\) for the
u\ and mi is admissible, then u\ is admissible
Lemma 3.9 Assume that u\ is admissible and that
admissible
second time is
2n. The
>
We say that mi is admissible if the
embedding
on
or
<
{u
to
an
u\
=
N(u\),
a
0}, i.e.,
M2
<
Mi
mi leads to
mi either leads to
Nn.
embedding
After
=
1, shrinking
3, shrinking
>
lead to
N(u\)
If N
=
an
In the second case,
after N
—
1 folds, i.e.,
contradiction.
mi, if necessary,
we may assume
shrinking
of
T4(A(a,
i.e.,
mi)),
right
boundary"
"upper
u^+i +n
even.
—
n,
since otherwise
fold another time at
D(n) into T4(A(a, u\))
u
N(u\)
=
with u^+2
u^
<
—
<
N
n
and obtain
—
1. Therefore m^
"A'-t-l +7r- As
we
an
>
27T,
embedding
have
seen
of
above,
3.3.
Embeddings
shrinking
mi leads to
Assume that
Lemma 3.9,
we see
G
a
an
49
admissible
u\
assume
2n.
>
By
optimal embedding
+ 2n. If N
a/2
P(7r, a)
a
u\+U2-\
=
n
=
2n
find that the
folding
N
—
2k
+
2k
=
—
1 and
by folding once
off from Figure 22
a
By
a+n,
better
a).
those
analyze
touches
Fjv+i
embeddings
{u
=
0}. The
by A$a)
=
that
+ un+i
2(u\ -n)+ 2(u\
-
3n) H
\-
2(u\
Nn) +
-
n
u\+2n,
Solving for mi and using the formula A (a, mi)
of
into
ball
obtainable
x
a
R(a)
D(n)
optimal embedding
by
1 times is described by
Nn.
=
—
—
2n
=
2k
r.
h
(k
+
provided
2)n
that
a
2(kl
>
-k+ $n.
1. Define the function
=
spß(a)
]2n, oo[ by
spb(o)
cf.
provide
is therefore described
Observe that this formula also holds true forN
on
+
>
+2ku\-2k2n
>
a
Ajk(fl)
read
=
provided that Mi
we
we
B4(n
<-h>-
Lemma 3.9 it suffices to
not
1.
=
a/2+2n
=
obtainable
3,
>
is admissible. Then N(u\)
mi
=
for which the number of folds is N
D
un+i +n.
=
a/2. Since A(a, a/2)
[n,2n] multiple symplectic folding does
that mi
result than the inclusion
assume a
u'N+2
such that
[n, 2n] and that
G
a
may
we
that for
embedding
So
into balls
Figure
=
23. We in
Proposition
min
| Ak(a) \k
particular
3.10 Assume
have
>
a
1,2,
=
...;
2n.
a
=
>
2(k2
—
k +
l)n
\,
proved
polydisc
Then the
embeds into the ball B4 (sps(a) + e)for every
spb(o)
a
—
e
a) symplectically
P (n,
>
0, where
+
2)n
2n
\-(k
——;
2k
for the unique integer kfor which
Remark 3.11 Let
dps(a)
and the volume condition.
2(k2
—
k +
increasing
=
sps(a)
—
—
The function
\)n, where dpß(ak)
sequence
2(k2
converging
—
to 2n.
(2k
k +
1)
\/2na
dps
+
<
a/n
<
2(k2
+ k +
1).
be the difference between spb
attains its local maxima at a^
\)n
—
2ny/k2
—
k + 1.
This is
=
an
O
50
3.
Extend the above function
a
+
n
for
fps
fps(a)
The
on
to a
[n, oo[ defined
inf
=
|A
function
[n, oo[ by setting spß(a)
on
3.12 The
=
understand
by
summarizes what
function fpß
max
was to
B4(A) \
| P(n, a) symplectically embeds into
following proposition
Proposition
above by
spB(a)
in four dimensions
[n, 2n]. The problem considered in this section
G
a
the function
Multiple symplectic folding
:
(27T, \/2na I
know about this function.
we
[n, oo[
R is bounded from below and
—>-
fpß(a)
<
.
spß(a),
<
Figure 23. It is monotone increasing and hence almost everywhere differen¬
tiable. Moreover, fpß is Lipschitz continuous with Lipschitz constant at most 2;
see
more
precisely,
fpß(a')
—
fpß(a)
(a'
<
—
for all a'
a)
> a
> n.
a
Proof.
ity,
In view of the
the identities
(1.5)
and
axiom for the second Ekeland-Hofer capac¬
(1.9) imply fpß(a)
<
<
Proposition
Assume
a
<
a'.
If cp
symplectically
P(n, a)
<P\p(n,a) symplectically
and so, as every increasing real function,
note the dilatation by a'/a by À. Assume
into B4(A). Then the composition
Pâ')
£i
symplectic embedding.
fpß(a')
-
fpB(a)
<
into
P(^,fl)
Therefore
fPß(a)
claimed.
n.
The volume
+/2na.
>
We
con¬
2n, the estimate fpß(a)
<
B4(a
<-^-
-
a
embeds
B4(A).
almost
that
+
n), and for
-
P(n,a')
everywhere
t B4(A) X
1
\
)
into
>
a
2n
<
<
—
B4(A),
Therefore fpß is
<
by
then
increasing,
differentiable. De¬
ij/ symplectically
fpß(a')
fa'
\
as
fpp(a)
>
a
3.10.
embeds
a
2n for all
to
a
spß(a) is provided by the inclusion P(7T, a)
is
>
\B4(fpß(a))\ translates
fpß(a). For
max(27r, \j2na)
condition \P(n,a)\
clude that
monotonicity
embeds
P(n, a)
B4(â)
fpß(a). We conclude that
spb(o)
,
-^^(a'
-
a)
,
<
2(a!
-
a)
a
D
3.3.
Embeddings
into balls
51
A
/A
+
a
=
n
a
1—i—i—i—i—i—i—i—i—i—-
n
Figure
For
of
a
23: What is known about
than the inclusion
ities do not
| P(n, a) |
lOn
2n, multiple symplectic folding does
=
fpß(a)
6n
2n
provide
<
a
P(n, a)
°->-
54(7r
better lower bound of
| B4(fpB (a)) |.
Question 3.13 fpß(2n)
We end this section
<
fpß(a).
provide a better upper bound
not
a), and Ekeland-Hofer
+
fpß(a)
We therefore would like to know the
answer to
3n ?
by deriving Proposition
3.8 from
Proposition
3.10.
spß
(|)
trapezoid T(a, n)
and
Proof of Proposition 3.8: Computer calculations suggest that S£ß(a)
for all
a
>
2n. For
Lemma 3.14 We have
Proof.
As in 3.3.1
of the ball
B4(A)
purpose, the
our
we
as
seb(o)
<
following
spß
think of the
the
(f
n) for all
4-
ellipsoid E(n, a)
trapezoid T4(A).
F
=
We fix
of the
a
a
the
as
>
2n.
>
2n and let
N
]jFi u\JSi
i=l
image
<
result will be sufficient.
N+l
be the
capac¬
than the volume condition
i=l
"optimal" embedding
of P
(n, |
4-
n)
into
T4
(spb (§
+
We recall from Lemma 3.9 that the number of folds N is odd and that for N
the set T looks
as
in
Figure
22. We define A
G
]27r, oo[
for which
seb(A)
=
spb
(-
+
7r)
as
the
unique real
tt)).
=
3
number
52
3.
and
we
Multiple symplectic folding
let
OO
*"
00
Lh/uLJ3
=
i=l
be the
ure
of the
image
i=l
"optimal" embedding
of
T(A, n) into T4 (seb(A)), cf. Fig¬
21.
In the
the
in four dimensions
sequel
embeddings
the stairs
TJ
we
shall compare the volume of !F with the volume of T'. Since
of P
Si and
(n, |
JJ
+
n)
and
T(A,n)
are
both
S- "vanish". We shall therefore
"optimal", the volumes of
neglect the stairs of both
sets.
Recall from 3.3.1 that
the width and the
/,- denotes the minimal height of the floor F! and that
of the i'th
height
T{
"triangle"
T4(sEB(A)) \
in
T' is 2/2,-1.
Also recall that
n
h
>
l2
>
>
(3.12)
....
Folding Lemma 3.1 imply that F, C 3T' for each odd / > 3, and
that F \ Jr' is the disjoint union of the thin "triangle" Qx
F\\ (F[ U F2), the
/
each
contained
in
different
a
C
F,-, even,
"rectangles" g,
triangle T'.,^, and the
set go lying in the left end of the floor F^+i, see Figure 24.
Using the definition (3.1) of l\ and the estimate (3.3) we find
This and the
=
n2
n
l2
and
<
l\
n
=
mi
A
<
n
—
A +
n
so
|gol
+
|gl|
<
/27T
+
-/2^/2
<
TT2-
(3.13)
We shall prove that
I02l
The estimates
+
l04l
+
---
+
lo*+il
<
|*"\^|-
(3.14)
(3.13) and (3.14) yield
|F|
l^x^'l + l^n^'l
w2 + |F/\Jr| + |5r/nF|
=
<
_2
n
=
I (7-/1
+ \J,
Therefore,
o
7ra
p(n,- + n\\
=
|F|
<
n2+\F'\
=
n2+\T(A,
n)\
=
n:
nA
3.3.
Embeddings
a
seb
(à)
<
<
A.
53
24: The sets
Figure
i.e.,
into balls
ß,-c5r\ F'
Since the function seb is monotone
spb
(|
+
n)
as
Qi
C
T'j{i),
i
whose width
-h(Qi),
w
we
(Ri) is equal
cf.
Figure
increasing,
denote the
we
conclude that
in
T4 (seb(A))\
C
=
?"
to the
n
Ti/2
c
2l2j(i)-i
-
w
(Ri)
=
rectangle
5" \ F
height h(Q{)
24. Since the width
"triangles"
and associate with each
1,2,...,
by Ti,
2, 4,..., N + 1, the rectangle
=
i
Ri
2n
(3.14)
and width 2n
height
Rj(i) Cf'\f'
claimed.
In order to prove the estimate
F of
and the sets
of
w(Qt)
2l2j(i)-\
g,- and whose height
of
-
g,-
h
is
(Qi)
/z
(/?,-) is
54
3.
find
we
together
with the
in
higher
dimensions
inequalities (3.12) that
\QA
=
w(Qi)HQi)
=
(2l2j{i)-i-h(Qi))h(Qi)
<
(2n-h(Qi))h(Qi)
h(Ri)w(R{)
=
\Ri\-
=
The estimate
Symplectic folding
(3.14) thus follows,
and
so
the
Proposition 3.8 follows from Lemma
Proposition 3.10 the function
proof of Lemma
3.14 and
3.14 is
Proposition
complete.
3.10.
Indeed, in
view of
d(a)
on
:=
]27T, oo[ has its local maxima
ak
2
=
(2(k2
spb
+
n)
—
*Jna~
at
k +
-
(|
1)
1
-
\n,
k=l,2,...,
where
d(aji)
The sequence
d(ai)
=
is monotone
(2k
+
\)n
—
^/na^.
increasing
to
2n.
(|
+
n)
Together with Lemma
3.14
we
conclude that
seb(o)
and
3.4
so
the
proof
of
s/na
Proposition
Embeddings
In this section
—
<
spb
3.8 is
—
s/na
<
2n
complete.
into cubes
multiple symplectic folding to construct symplectic embed¬
dings of four dimensional ellipsoids and polydiscs into four dimensional cubes.
While in Section 1.3 the lack of convenient invariants made it impossible to get
good rigidity results for embeddings into cubes, multiple symplectic folding pro¬
vides us with rather satisfactory flexibility results.
we use
3.4.
Embeddings
into cubes
A(a, uq).
uq
25: The
Figure
3.4.1
Fix
x
G
<
an
embedding
we are
the
<
m
,
w
(a,Mj)
/
>
w(a,u\)}.
of
the
By
=
Folding
Ml+/l
condition for
=
If the remainder r\
a
7T
we
T(a,n)
proceed
Lemma 3.1
+ I1
2, maximal with respect
T(a, n) into {0
forced to fold
only
In.
=
as
C4(A)
embeds
follows. We
(i), the stairs Si
are
w(a, u\)}, where
<
We then choose m;
{0
a
T(a,n) and of the cube
as
D(A) via multiple symplectic folding,
{u
(A) for
C
cubes
ellipsoid E(n,a)
]0, a[ and fold at u\.
contained in
c-»
In order to find the smallest A for which
D(A).
x
R(A)
fix mi
We think of the
n.
>
a
into
optimal embedding F(7r, a)
Embedding ellipsoids into
R(A)
as
A(a, mq)

u\,
in 3.3.1 shows that
i.e.,
u\,
an
a
If this condition is met,
1, 2,
...,
>
into
{0
+
n
define M2
of the widths of the stairs
conditions at the
all i
we
(3.16)
>
u\
subsequent folds,
by formula (3.4). The sequence /,-, i
S( is decreasing. Hence, there are no further
=
and if there is
3. Under the condition
an
i'th
fold, then
U{
>
m(_i for
(3.16),
T(a,n)
procedure
w(a,u\)} by folding finitely many times. We denote the number
of folds needed by N(u\). Recall that /,•
/,• (a, u\ ) is the width of the stairs S(- as
<
m
therefore embeds
our
<
=
well
i
—
as
1,
the minimal
...,
N(u\).
height
of the floor F,
We set
U(a, u\)
=
as
well
0 if i
>
as
the maximal
N(u\).
For
a
height
>
n
of
F;+i,
fixed and
56
mi
3.
G
]^jr, a[,
the functions
k(a,u\)
Symplectic folding
decreasing
are
in
higher
dimensions
continuous functions of mi.
Therefore, the height of the image
tf(«i)
h(a, mi)
^
=
k(a, mi)
1=1
is
a
decreasing
have
h(a, u\)
/;(a, mi)
->
continuous function of u\ for mi
—
and for mi \ an/(a + n)
n,
for all i
/i(a, Mi)
On the other hand,
u>(a, mi)
function of mi such that
h(a,u\)
for
function
SEc(a)
exactly
>
2, and
tt
+
=
w(a, 0)
one
mi
Figure
C4(sec(o) )
sEC'(a) is presented in Appendix D.2.
cube smaller than
an
is
procedure
+
for any
optimal
in the
be described
Then
as
sec(<z)
respect
struction and for which the
improving
A
< s EC
or
our
procedure
(a), and choose
the
and hence
which
—
oo
a/2
and m
=
It follows that
a.
call mo
we
mo
=
we
—>•
w(a,u\)
0,
(a).
=
Define the
j Mo(a),
ellipsoid F(7T, a) symplectically
>
0. A
sense
that
For each A
we
cannot embed
of the
height
larger than
m-
G
height
of the
image
<
image
This follows from
run
we
into
some
embedding ob¬
A. The only way
an
is smaller than
of the m/ smaller. So let
m,-. We then either run into an
of the modified
a
procedure can equivsuccessively choose
staying inside {0 < u < A).
do not
is therefore to choose
E(n, a) into
our
]n, a[
to the condition of
we
embeds
computer program for the function
in 3.3.1. Indeed,
is the smallest A for which
of
tion,
e
given
one
follows:
maximal with
u\, M2,...
N(u\)
C4(sec(<*)) by multiple symplectic folding.
argument similar to the
alently
iu(a, a)
+ M
n
=
26. We have shown that the
into the cube
Our
have
>
]n, oo[ by
on
SEc(a)
cf.
and
n
=
we
For mi
so /?(a, mi) —> oo as mj \ an /(a + n).
(\— n/a)u\ is a strictly increasing continuous
]^^r, a[,
g
]^t^t, a[.
g
embedding
is
embedding obstruc¬
larger than h(a,u\)
A.
Remarks 3.15
1. We
are
We have
going
to
investigate
=
and
only
so
if u\
sec(®)
in
more
detail.
n < n +
[a/2, a[. Then h(a, u\)
and
<
w(a,u\)
h(a, u\)
(1 —n/a)u\
w(a,u\),
w(a, a/2)
(a+n)/2,
for all mi G [a/2, a[. It follows that by folding once we can embed F(7r, a) into
C4 {^y~ + e) for any e > 0, and that folding only once never yields an optimal
embedding, i.e., sec(ü) < (a + n)/2 for all a > n.
N(u\)
=
1 if and
the function
G
=
=
3.4.
Embeddings
We have
into cubes
N(u\)
57
2 if and
=
if mi
only
a/2 and I2 +
<
w(a, mi). Using the formulas (3.15), (3.1) and (3.4) for
the second
inequality
is
w,
U3
l2
=
h and
+
M2,
(a/n)l2
we
<
find that
to the condition on mi
equivalent
a(a2
n2)
+
3az + nl
If
N(u\)
the
2, then h(a, u\)
=
identity
a
=
u\ + U2 + U3
=
2l\ + l2. Plugging the identity
u\ + U2 +
=
(a/n)l2
2n
h(a, mi)
2n
=
mi +
n(a
a
The
h
equation
w
=
thus
that
Mo(a)
meets condition
an(2a
a/2 whenever
(3.16),
compute that
We
n, and that
Mo(a)
<
if
(3.18)
fact, the identity (3.18) also holds
< a
n
In
<
>
a
3n. It follows that
mal
mi
l2
I2 into
—
we
find
n
—
of
n)
—
~1-^
aL + 2an
=
meets condition
(3.17).
solving
=
for
2mi)
—
a
M2
yields
"0(a)
provided
and
that
Mo(a)
Mo(a)
(3.18)
—2
—
nÂ
Mo(a)
meets
meets
holds for all
a
a/2,
<
condition
and that
Mo(a)
(3.16) and that
condition (3.17) if and
G
true for all
only
]n, 3n].
those
for which the
a
opti¬
T(a, n)
embedding
by multiple folding
2l\ + I2, i.e., for which M4(Mo(a)) < M3(Mo(a)). The
height is still h
largest a for which (3.18) holds true is characterized by the identity M4(Mo(a))
M3(Mo(a)). Using the identity M3
fz^r"2 we compute that the equation a
the
obtainable
is
a
3-fold for which
=
=
=
=
uo(a) + u2(uo(a)) + 2u3(uo(a)) translates into
an(5a2
a
(a
i.e.,
a
=
(2 +
\/5j
n.
an(3a
In
a
general, sEc(a)
for which
=
—
n)(a2
—
+ 2an
n)
-=—
~
az + 2a7r
is
a
u^â)(uo(a))
for which the
N(a)
=
2an +
n2)
—
n2)
'
Therefore,
/
secW
—
=
"endpoint"
—
piecewise
=
of
for
n
< a
<
7rz
/-.
t
V5)7r.
(2 +
rational function. Its
singularities
are
those
3,5,1,..., and those a
u^â)+\(uo(a)), N(a)
touches
the
"axis"
{u
0}; here, we set
F^(a)+i
N(uo(a)). Denoting the
—
=
two sets
of
singularities by
03, as, a-j,...
and
58
3.
04, ae, a%,...,
verging
we
(aii),k
sequence
2. Let
dEc(a)
volume condition.
tion
dEc
attains
dsc attains its
have that the
=
s
>
exactly
=
local minima m^ at a2k+i, and that
analyze
dsc strictly increases be¬
to
they suggest that both (m^) and (M^)
(2/3)n. In particular, we seem to have
O
=
of the function seb studied in 3.3.1
asymptotic
di¬
1. Moreover,
>
-
case
strictly increasing,
*Jna/2 be the difference between sec and the
n.
Computer calculations suggest that the func¬
strictly increasing and converging
limôo sEc(a)
+Jna/2
(2/3)n.
the
the
dimensions
—
are
As in the
set of sec is
higher
local maximum M^ between a2k and a2k+i, that
one
and a2£+2> k
tween a2£+i
singular
in
3.
EC(a)
Set a2
Symplectic folding
behaviour of the function
we
sec(<*)
do not know how to
as
a
-»
oo
directly.
by
following proposition
subsequent
comparing the optimal multiple folding embedding of F(7r, a) into a cube with
the optimal multiple folding embedding of the polydisc P (n,
| + 7r) into a cube.
We shall prove the
Proposition
at
the end of the
section
3.16 We have
Ina
SEc(a)
—
\
~z~
3
far all
~7T
—
a
>
In
A
4n
A
A
—
-
a+n
~1T
3n
2n
n
2n
Figure
3n
4n
5n
6n
In
26: What is known about
fEc(a)-
.
3.4.
into cubes
Embeddings
Extend the function
problem
defined
function
was
Proposition
inf
=
I
A
3.17 The
function
fEc
n.
=
on
The
\n, oo[
summarizes what
we
function fEC'- [^» oo[
C4(A) \.
know about this function.
R is bounded from below and
—>
by
Figure 26.
It is monotone
tiable. Moreover,
more
[n, oo[ by setting sec(k)
| E(n, a) symplectically embeds into
n\ax(n,J—\
see
on
to understand the
by
following proposition
above
to a
considered in this section
/gC(a)
The
sec(o)
59
is
fßc
fEc(a)
<
increasing and
Lipschitz
sEc(a),
<
hence almost
everywhere differen¬
Lipschitz constant at most 1;
continuous with
precisely,
fEc(a)
—
fEc(a)
SEc(a)
<
(a —a)
for
all
>a>n.
a
a
In view of the
monotonicity axiom for the first Ekeland-Hofer capacity,
(1.5)
(1.9) imply fEc(a) > n for all a >n, and the volume con¬
dition \E(n, a)| < |C4(/ec(a))| translates to fEc(a) > y/na/2. The first claim
thus follows. The remaining claims follow as in the proof of Proposition 3.12.
Proof.
the identities
3.4.2
Fix
a
C4(A)
and
Embedding polydiscs
>
as
n.
We think of the
R(A)
x
D(A).
R(A)
We fix mi
]0, a[ and fold
g
contained in
{u
the condition of
The
Folding
<
D(A)
u\ +
staying
mi
n}. We then choose
inside
>
{0
(ii)
n.
<
For N
>

A(a, mi)
=
m(-, i
>
2, maximal with respect
to
n}.
only condition for folding a second
only condition on mi for folding an
n.
procedure
finite number
u\ +
shows that the
We say that mi is admissible if mi
D(A(a, mi)) by
R(a)
multiple symplectic folding, we proceed as follows.
at mi. By the Folding Lemma 3.1 (i), the stairs Si are
.A/'th time, if necessary, is mi
u\ is admissible, then
as
via
Lemma 3.1
time, if necessary, is
polydisc P(7r, a)
In order to find the smallest A for which
embeds into
x
into cubes
>
a/2
embeds
N(u\)
or
R(a)
>
mj
x
n.
D(n)
It follows that if
into
R(A(a, u\))
x
of folds. Here,
max{Mi +n, (N(u\)
+
l)n}.
(3.19)
60
3.
uiui +
Figure
Let M2,
«a/(Mi)+i be the
of the m,, i
2,...
ing some
folding at
...,
embedding
G
a
also
]n, 2n]. Then A(a, mi)
2n.
>
+
an
u\
P(n, a)
that
F^+i
that
however,
touches
by Ai(a)
=
Fjv(M])+i
max
read off from
{|
+ n,
Figure
that u\
for mi and
D(n) into
using
a
>
we
find
=
mi +
{m
u\
2n)
does not touch
<
u\,
{u
|
=
2n,
<
a
n).
=
obtainable
provide
not
a
N(u[)
a
u'x as
{u
=
0}
It
better
N(u\) is
:=
N and
F'N+1
claimed
or to
=
n),
u\ +
then
touches
u'x
n.
=
contradiction. We may therefore
A similar argument shows that
if
a
we
may
is odd.
N(ui)
by folding only
once
is therefore described
+ n, and if the number of folds is N
>
2,
we
27 that
a
provided
Choos¬
C4(a).
°->-
mi either leads to
touches
optimal embedding
F/v+i
such that
n}. Indeed, shrinking
assume
The
.
27r for every admissible u\.
>
that mi is admissible and that N
Suppose
admissible
In the second case,
assume
admissible mi
some
]n, 2n] multiple symplectic folding does
G
a
assume a
j
associated with
lengths
result than the inclusion
there exists
{m
three times.
Folding P(n,a)
We claim that if the last floor
m
a
procedure.
our
follows that for
=
dimensions
,N(u\), smaller would lead to an embedding by
N(u\) times. It is therefore enough to compare embeddings ob¬
least
Assume that
even.
n
higher
=
tained from
So
27:
in
Symplectic folding
n.
Since
formula
AN(a)
>
a
(3.19)
cube obtainable
2n +
=
we
(N
2n,
l)(u\
we see
max
n)
-
that this condition is met.
then find that the
by folding
=
+
optimal embedding
by
N times is described
{s^ZL, (Ar
+
i)7r
j
Solving
of
R(a)
x
3.4.
Embeddings
into cubes
61
Observe that this formula also holds true forN= 1. Define the function spc(a)
on
]27T, oo[ by
spc(a)
cf.
Figure
54. We in
Proposition
particular
3.18 Assume
min
{AN(a) \
have
proved
=
a
2n.
>
embeds into the cube
C4(spc(a)
1
(N+l)7t
SPC{
The function
_
+
Then the
e)for
every
1, 2, ...},
=
polydisc P(n,a) symplectically
e
if (N-\)N
if N2+\ <
^fr
I
N
>
0, where
+ 2
%
§
<
<
N2
N(N+l)
<
+
l,
+ 2.
spc(a) is compared with the result yielded by Lagrangian folding in
6.2.2.1.
We end this section
by deriving Proposition
Proof of Proposition 3.16:
Lemma 3.19 We have
Proof.
As in 3.4.1
fix
2n and let
a
>
we
proceed
We
sec(<z)
<
(f
spc
think of the
as
in the
+
ellipsoid E(n, a)
For TV
image
=
of the
a
as
>
the
3.18.
3.8.
2n.
trapezoid T(a, n).
We
N
IiF< ULI5<
=
i=l
be the
Proposition
proof of Proposition
x) for all
AH-l
T
3.16 from
i=\
"optimal" embedding of P (n, | + n) into C4 (spc (| + n)).
as in Figure 27. We define A G ]27r, oo[ as the unique
3 the set S7 looks
real number for which
sec(A)
and
we
spc
=
(- + nj
let
A"+l
r
=
A"
]Jf/u]Jsj
(=i
be the
image
of the
(=i
"optimal" embedding of
F
(A, n) into C4 (sec(A)), cf. Figure
25.
As in the
proof of Lemma 3.14
from 3.4.1 that
we
shall
neglect the
stairs of !F and 5r/. Recall
/(- denotes the minimal height of the floor
n
>
l\
>
l2
>
....
F/
and that
62
3.
This and the
Folding
Lemma 3.1
\ F'
T
gi := Fi \ F'
has volume
24,
Figure
The set
imply
(F{
lÖil
decompose
We
the set
go
higher
dimensions
FN+i \ 3=".
U
U
FÇ),
=
-h
which is
—
2
F^+i \ !F'
:=
in
that
Fi \ F'
=
F\ \
=
Symplectic folding
A
analogous
to the set
gi in
h-
into the sets
Q'0'
and
g0
=
g0 \ g0,
where
Qo--=
Using
{(u,
v, x,
y)
G
go |
m
<
l\)
{(m,
v, x,
y)
G
go |
m
>
sec(A)
the definition
(3.1) of l\ and the estimate (3.16)
<
l\
=
n
A
odd,
if N is
even.
find
we
n2
tt
l2
and
h)
—
if N is
<
mi
n
—
A +
n
so
|ßo|
+
löil
<
hit + -l2-l2
it2.
<
(3.20)
2h~A
We shall prove that
|öo|
The estimates
3.14
that
yield
(a)
s EC
We
sec(A)
are
—
the floor
Qq
(3.20) and (3.21) and the
<
a
<
spc
(f
+
F'N,+V
j
n)
same
(3-21)
argument
assume
the estimate
first that N' is
even.
length of F'N,+l is
(sec(A) —In1)- Therefore,
<
(sec(A)
Let R be the union of maximal
"rectangles"
-n
< m
<
sec(A)
(sEC(A)
in 5"'
length
Recall that
at most
-n-h)j
{(m, v) I sec(A)
The
(3.21).
Since the
| ßo'l
in the
proof of Lemma
increasing, we conclude
as
claimed.
as
proving
—l\. We
is at most
|^"\^|.
A. Since the function sec is monotone
left with
n
<
\
-
l^<
—
over
sec(A) -l\).
is the
In', the
lNr).
F based
g0;
is
height
height
of
of the set
of
(3.22)
3.4.
Embeddings
into cubes
63
If N is odd, R has
one
component, whose height is
even, R has
two
components, whose total
Together
one or
with
n
-
|S" \ F\
Since
l\
>
so
=
1*1
>
(sEc(A)
j
>
(3.21) follows.
In' replaced by In'-i
the
-
h)
(sec(A)
j
proof of Lemma
Assume
Proposition 3.16 follows from Lemma
view of Proposition 3.18 the function
on
h) (sec(A)
The estimate
through.
complete.
d(a)
]27r, oo[ has its local maxima
aN
=
2
(n2
spC(l
:=
-
In'-
If N is
at least sec (A)—n
—
In'
lN.).
-n-
one
in
(3.23)
(3.22), and
so
that N' is odd. Then the above argument
now
goes
3.19 is
—
conclude that
we
-
height is
In', the right hand side in (3.23) is larger than the
the estimate
with
h
least sec(A)
at
3.19 and
+
n)
(3.21) is thus proved, and
Proposition
3.18.
Indeed, in
/7ra
-
at
N +
l)
iV
n,
=
1, 2
where
d(aN)
The sequence
we
d(aN) is
=
(N
-
VN2
increasing
monotone
-
N +
\n.
to
l)
n.
Together
with Lemma 3.19
conclude that
,
x
\7ta
In
SEc(a)-^—
and
+ 1
so
the
proof
of
Proposition
<
spcii
3.16 is
+
\
llta
Jtj-^Y
complete.
3
-
1*
64
3.
Symplectic folding
in
higher
dimensions
4
Symplectic folding
in
higher
dimensions
Even
though symplectic folding is a four dimensional process, we can use
prove interesting symplectic embedding results in higher dimensions as well.
reason
is that
we can
dimensional fiber
on
fold into
over
n
—
embedding skinny polydiscs
chapter
4.1
types of folding
In
Chapter
3
folded
we
the
on
multiple folding procedures
(—y )-direction.
be chosen
Chapter
and
right
skinny ellipsoids
the left into the
on
into balls. The
5 to prove Theorem 3.
considered in this
y-direction.
In the
shall also fold into the
chapter we
types of folding. This section reviews
neglect those terms in the constructions which
Hence there will be four
these four types. As
can
—
into cubes and
will be used in
The
symplectic directions of the (2n 2)symplectic base. We will concentrate
the 2-dimensional
results of this
Four
1 different
it to
usual,
we
shall
small.
arbitrarily
R4 by
We define F C
F
Fix mi
{ (m,
:=
v, x,
y)
R and choose
G
R4 |
G
u G
R, 0
[Mi,Mi+7r]anda (left-)cut off function
1.
Folding
We fold F
on
the
on
the
right
right
into the
at
u
=
v
1, 0
<
off function cr
(right-)cut
a
<
R
q :
—
< x
:
<
R
—>
1, 0
<
y
< n
}.
[0, 1] with support
[0, 1 ] with support [m i
—
n,
u\
].
y-direction
u\ into the
y-direction by applying
the
symplectic
map
0r+
:=
(y\
Here, the maps ß\ and y\, which
and y in
Step
1 and
Step
x
id)
are
o
<p\
o
(ß\
x
id).
constructed the
4 of Section
2.2,
are
same
explained
way
as
=
I u,x,
v
+
cr(u)x,
y +
j
ß
in the first column of
Figure 28, and the lifting map <p\ is defined by
cp\(u,v,x,y)
the maps
cr(s)ds\.
66
4.
Symplectic folding
in
higher
dimensions
1
0.
101
i
>
M
—-
M
104
n:
1||
I
jjl
u\
u\+n
u\— n u\
u\+n
u\
lj
mi
n
mi
n
u\
—
1
0
K3
1.
0
mi +
mi
n
u\—
Figure
2.
Folding
We fold F
on
on
n
28: The maps
the left into the
the left at
u
—
</>/+
Here, the maps 02 and y2
u\
ßi and
into the
x
=
1, 2, 3, 4.
o
cp2
o
(yÖ2
x
3.
Folding
We fold F
on
on
the
the
y)
v, x,
right
right
into the
at
u
</v_
Here, the maps 03 and y?
cp3(u,
4.
Folding
We fold F
on
on
v, x,
\u,x,v
=
in the second column of
y)
(y3
:=
I m, x,
the left into the
the left at
u
0/_
=
x
id)
explained
=
v
y +
/
s)ds\
ci(s)ci(ds
)
and
.
(—y)-direction by applying
o
<p3
o
(ß3
x
cr(u)x,
y
—
the map
id).
in the third column of
—
Figure 28,
and
J cr(s)ds\.
(—y )-direction
u\ into the
:=
ci(u)x,
Figure 28,
(—y )-direction
mi into the
=
are
—
the map
rJ).
/
<P2(u,
—
y-direction by applying
id)
explained
are
y;, /
u\
y-direction
(72
:=
mi + n
mi
u\
(y4
x
(—y)-direction by applying the
id)
ocpô
(ß^
x
id).
map
Embedding polydiscs
4.2.
Here, the maps 04 and y4
into cubes
4.2
explained
are
<p$u,v,x,y)
I m, x,
=
Embedding polydiscs
In this section
we
shall
P2n(n,
into cubes
shapes.
C2n (A) for
Define the
...,n,a)
n
2.
>
we
D(n)
=
before,
{(x, y) |
—
abbreviate
Rn(a)
to
0
x
•
y
x
•
we
< x
R(a, 1)
R(a, 1)
=
D(a) is symplectomorphic
I
—
ci(s) ds \
.
of
skinny polydiscs
D(n)
x
D(a)
shall work with
more
convenient
=
x
x
0
<
R(\, b)
x
<
a,
R(l, b)
Rn(a, a).
x
•
•
x
y
b\.
<
x
R(l, b) by
R(l,b).
In view of
Lemma2.5, the disc
R(a, 1) and the disc D(n) is symplectomorphic
to
polydisc P2n(n,..., n, a),
symplectomorphic to
to
Rn(a,
n).
symplectomorphic
Similarly, the cube C2n(A)
symplectomorphic to Rn(A). The symplectic coordinates will be denoted by
R(\, n).
P2n(a,
is
ci(u)x,
and
rectangle R(a,b) by
Rn(a, b)
a,
+
Figure 28,
into cubes
As
We denote the 2n-dimensional set
—
v
in the fourth column of
study symplectic embeddings
R(a, b)
If b
67
where
Therefore, the
n,...,
we
set
U'2» ,yn)
n),
is
(m,
v, x2, y2,..., xn,
again (u,v)
as
which is
well
yn)
=
(zi,z2,..., zn)
(x\ ,y$. We abbreviate
=
x
=
e
R2n
(X2,..., xn) and
y
=
as
\y
=
(1,...,1)
We shall associate with each
triple
a
G
>
R""1^).
2n, N
e
¥i,
n
>
2
a
symplectic
embedding procedure
<pnN(a): Rn(a,n)
In the
ing
following description
we
in the actual construction.
again neglect
<-+
the
R2n.
arbitrarily
small 5-terms appear¬
68
4.
If
n
=
define mi
2,
we proceed
u2 N(a) by
=
a
Then mi
>
in
as
3.4.2, cf. Figure 27: For
2n +
=
R2(a,
n.We first fold
Symplectic folding
n)
a
in
>
higher
dimensions
2n and JVgNwe
(N +\)(ui -n).
on
the
(4.1)
into the
right
y2-direction by applying
the map
(Zl,Z2)
at
m
mi.
=
4.1.1. If N
Here, <pr+ is the restriction
image
implies
that
on
hyperplane {m
v, x2,
y2)
0r+
G
the
m
0}. If
the left into the
[R2(a, n)\ \
second floor at
H>
>
2, the inequality
>
u\
n
u
<
u\, y2
> n
\
the map
Cpl+(Z\,Z2)
Going on this way, we altogether fold N times into
y2-direction by alternatingly folding the last floor of the image on the right at
=
m
n.
=
mi and on the left at
u
n.
=
At this
terminates. Indeed, in view of definition
the last floor of the
the
point
embedding procedure 4>N(a)
the
(4.1), "a is used up" and the front face of
image touches the hyperplane {m
0} if N is odd.
hyperplane {u
If n
3, the embedding procedure (pN(a) can
> 2n and iVeNwe define mi
u\ N(a) by
—
u\ +
n) if
N is
even
and
=
—
a
N
y2-direction by applying
(Zl,Z2)
to the
=
then fold the second floor
we can
image
n) of the map cf>r+ introduced in
(pN(a) terminates at this point. Indeed,
(4.1), "a is used up" and the front face of the second floor
touches the
(m,
of the
R2(a,
to
1, the embedding procedure
=
in view of definition
of the
<Pr+(Zl,Z2)
I-+
be visualized
by Figure
29. For
=
a
Then u\
>
n.
Set a'
=
=
2n +
ding procedure (pN(a) yield
restriction to
R?(a', n)
cp2N(a')
We next fold
once
x
a
2n +
(N +
(N + l)(u\
1)2(mi
it).
-
(4.2)
n). The first N folds of the embed¬
symplectic embedding of R3(a, n) into R6 whose
—
is
id:
into the
R2(a',n)
x
y3-direction.
R(\,n)
^
R4
If N is even,
we
x
R(l,n).
do this
by applying
map
(z\,
Z2,
zi)
h>
(z\,z2, z3)
where
(z[, z'3)
=
(pr+(zi, zi) and
z'2
=
z2
the
4.2.
Embedding polydiscs
into cubes
*-
Figure
N
29:
The first 5 folds of
an
embedding <pN(a):
R
R2n
(a, n)
with
4.
=
to the N + l'st floor of the
applying
to the N + l'st
part of the
using
image
at
m
=
u\,
and if N is odd,
we
do this
by
the map
(zi,z2, zi)
If N
M
u\ + n
Mi
i->
(zi, z'2, z'3)
floor of the
image
on
where
image
which y3
>
n
at
m
(z'i,z'3)
=
exactly
n, see
4>i+(zi,Z3)
and z2
Figure
We then fold the
29.
N times into the
restrictions of the maps </v_ and 0/_ and
1, the
=
thereby
fill
a
=
z2
(—y2)-direction by
second
zi-Z2-layer.
embedding procedure <pN(a)
point. Indeed, in
view of definition (4.2), "a is used up" and the front face of the last floor of the
0}. If N > 2, we fold a second time into the
image touches the hyperplane {m
y3-direction, and fill a third zi-Z2-layer. Going on this way, we altogether fold
1 times, in which we fold N times into the y3-direction, and thereby
(N + l)2
fill N + 1 zi-Z2-layers. At this point the embedding procedure (p3N(a) terminates.
Indeed, in view of definition (4.2), "a is used up" and the front face of the last
floor of the image touches the hyperplane {u
mi + n] if N is even and the
0} if A^ is odd.
hyperplane {m
In order to describe the embedding procedure 4>N(a) for n > 4, we proceed
by induction and assume that we have described the symplectic embeddings
=
terminates at this
=
—
=
=
cpN-\a'):Rn-\a',n)
->
R2""2,
>
27T.
70
4.
We define mi
m"
=
> n.
Seta'
2n +
=
2n +
=
(N+l)n-l(u]
l)n-2(Ul -n).
(N +
the
embedding procedure (pN(a) yield
R2n whose restriction to Rn(a', n) is
4>%~\a')
We next fold
xid:
Rn-l(a',n)
into the
once
in
dimensions
higher
N(a) by
a
ThenMi
Symplectic folding
a
x
yn-direction.
-n).
(4.3)
(N+l)n~2-l
The first
symplectic embedding
R(\,n)
R2n~2
->
If N is even,
x
of
Rn(a, n)
into
R(\, n).
do this
we
folds of
the
by applying
map
(Zl, ...,Zn)
to the last
(Z^ ...,Zn),
H*
floor of the
image
(z'i,z'n)
at
m
=
=
c/)r+(zi,Zn),z'i
mi, and if
N is odd,
i
Zi,
=
=
do this
we
2, ...,n-\,
by applying
the map
(z\, ...,zn)
(z\,..., z'n),
i->-
(z\,z'n)
to the last floor of the
=
z\=Zi,
cpi+(zi,zn),
i
=2,...,n—l,
n. We then fill a second z\
image at u
z„_i-layer
1 times.
by folding the part of the image on which yn > n exactly (N + l)w~2
If N
1, the embedding procedure (pN(a) terminates at this point in view of
definition (4.3). If N > 2, we fold a second time into the y„-direction, and fill a
third zi
1
Zrc-i-layer. Going on this way, we altogether fold (Af + \)n~l
=
—
=
—
times, in which
we
Zn-i-layers.
zi
view of definition
The
At this
the
4.1 Assume
a
>
embedding procedure 0^(a)
(Af+1)f=^=r
Proof. Fix
described
a—2:
a
>
+
2n. Then the
3.18 to
polydisc P2n(n,
C2n(s2pnc(a) + e)for every
(N+l)n,
=
thereby
fill N + 1
terminates in
(4.3).
embeds into the cube
?2"c(a)
point
following proposition generalizes Proposition
Proposition
cally
fold N times into the yn -direction, and
2n,
(N
(N
-
-
l)Nn~l
l)(N
+
e
f-2<
I)""1
f
<
(N
-
2
2n and iVeN. We define mi
embedding procedure yields
cj)N(a): Rn(a,n)
a
...
,n,
-
<
Rn(AN(a)
+
a) symplecti¬
1)(N+1)
N(N +
by equation (4.3).
symplectic embedding
^
dimension.
0, where
>
<
arbitrary
e)
The
l)""1
previously
Embedding polydiscs
4.2.
into cubes
71
where
An(o)
Solving equation (4.3)
for mi
An(o)
Optimizing the
symplectically
spnc(a)
choice of N
is defined
completes
we
find that
a
{
+
\)n}.
2n
,
(N+l)
N,
G
—
+
2n, (N + \)n
conclude that the
we
C2n
embeds into the cube
(s2pnc(a)
+
polydisc P2n(n,
e)
for any
e
>
,n,a)
0, where
...
by
s2pnc(a)
This
max
max
=
{mi +n,(N
:=
the
=
min
{AN(a) \N=l,2,...}.
proof of Proposition 4.1.
D
Remarks 4.2
1.
Arguing as in 3.4.2 we see that for a G ]n, 2n] multiple symplectic folding
provide a better embedding result then the inclusion P2n (n,..., n, a) °->
C2n(a), and that the procedure proving Proposition 4.1 is optimal in the sense
that we cannot embed P2n(n,... ,n,a) into a cube smaller than C2n (s2pnc(aj) by
does not
multiple symplectic folding.
2.
The functions
Lagrangian folding
In view of the
completed
spnc(a),
n
3,
>
are
compared
with the results
yielded by
in 6.2.2.2.
proof
in Section 5.1,
O
of the second statement in Theorem
we
also prove
Proposition 4.3 Fixn > 2. For every
and a symplectic embedding
cpa
:
3, which will be
Rn(a, n)
a
^
>
3n there exists
a
natural number N (a)
Rn ((N(a) + l)n)
such that the following assertions hold.
(i)
Ifu<n,
cpa(u, v,x,y)
and
if u
>
a
—
-
(u, v,x,y),
n,
<Pa(u,
v, x,
y)
=
(u
—
a
+
(N(a) + \)n,
v, x, y +
N(a)n\y)
.
72
4.
(u)
\cpa(Rn(a,n))\
hm
Proof.
Fix
aN
2n +
=
We recall that in the
have
the
neglected
Define 8n
0
>
2N. We set mi
G
1)m_1(mi
(N +
2
>
arbitrarily
in such
appearing
3((N+l)n~l
a
28n
—
\)n.
-
we
in the actual construction.
at
>
n
u
way that the
u\
=
—
"
In the actual construction
28n if i is odd and
8n- After
R"(aN, tt)
as
follows. We
atu
=
n
+
28n
of the part of the last floor which is
M-length
to
cpnN (aN, SN)
1)
28n-
+
equal
symplectic embedding
a
-
therefore achieve the i'th fold
we can
image
to the i'th stairs is
mapped
thereby obtain
l)"_1(Ar
2n + (N +
=
small .5-terms
find that mi
we
fold the last floor of the
even
n)
-
Nn and
=
by
associated with aN and N
if i is
dimensions
previous description of the symplectic embedding cpN (un)
8n
Since N
higher
l)n)\
+
2 and N
>
n
in
1.
=
\Rn((N(a)
aôo
Symplectic folding
folding (A^
+
l)n_1
1 times
—
we
Rn ((N + l)n))
^
where
aN
=
2(n
=
2n +
=
n
+
28N)
(N +
(N +
+
We abbreviate
\//n
inequality
28n
n
+
=
<
+
(N
l)""1 (mi
+
<PN
u\
-
28n
we
(u,v,x,y)
y)
=
(u
-
\)n~x
-
+
l)""1
-
\)8N
l) 8N
(4.4)
In view of the construction of
=
v,x,
((N
+
i^w and the
have
\jfN(u,v,x,y)
^n(u,
48N)
-
\)n.
{^N, $n)-
—
n
-n)-3((N+
\)n-\ux
\)n~l(N
-
aN +
if
(N
+
u
< n
+
l)n,v,x,y
if
u
>
un
(4.5)
28n,
+
—
Nnly)
n
—
28n-
(4.6)
Notice that
un
<
ajv+2
The function A''
:
for every Af
]n, oo[
->
N(a)
G
2N
and
un
—>
oo as
Af
—>•
oo.
(4.7)
N,
:=
min{
A^
G
2N
|
aN
> a
},
(4.8)
4.2.
Embedding polydiscs
into cubes
is therefore well-defined, and
a
symplectic embedding ßa
and the translation
the
case a <
(u, v)
:
N(ün)
R(a)
(u
\-^-
N. Fix
—
3n. Since
which is the
v)
a,
>
a
R(aN(a))
°->-
on
{u
>
a
—
<
a
aN(a),
we
find
identity on {u < n}
n], cf. Figure 7 for
symplectic embedding
Rn(a, n)
:
cpa
=
+ aN(a)
We define the
aN(a)
73
^
Rn ((N(a) + \)n)
by
(Pa
In view of the formulae
assertion
verify
\cpa(Rn(a,n))\
a
3n.
>
for all
a
X
id2n-2).
and in view of its
=
(ii),
assertion
we
definition,
cpa meets
\Rn(a,n)\
first of all observe that
and
<pa
(Rn(a, n))
3n.
Assume
we
we can
have
G
a
l)n)
a
=
now
(N(a) + l)nn
\Rn((N(a) + l)n)\
that
a
>
a2.
In view of
]aN(a)-2, a^(a)]. Using
(AT(a) + l)"7r
(N(a)
aN(a)-2
>
(4.7) and the definition
this and the formula
-
l)n-l(N(a)
-
3)
=
—
(4.4) for
+ 1
.
(N(a)+l)n7r
(N(a)
+
\)nn
(4.10)
(4.8) and (4.7) imply that
N(a)
Combining
+
further estimate
a
The definition
Rn((N(a)
\Rn(a,n)\
\Rn((N(a) + l)n)\
>
C
Therefore,
=
(4.8) of Af(a)
aN(a)-2
(0a
(4.5) and (4.6)
\<pa(Rn(a,n))\
>
~
°
(i) in Proposition 4.3.
In order to
for all
tN{a)
=
the estimates
—>•
oo
(4.9) and (4.10)
as
a —>
we
oo.
(4.11)
therefore conclude that
\cpa(Rn(a,n))\
aô\Rn((N(a)
and
so
the
proof
of
Proposition
4.3 is
=
+
\)n)\
complete.
D
74
4.
4.3
Embedding ellipsoids into
In this section
shall
study
Symplectic folding
in
higher
dimensions
balls
problem closely
related to
symplectically embed¬
B2n(A). As in Chapter 2 we
,n,a)
ding skinny ellipsoids E2n(n,
start with replacing these sets by symplectomorphic sets which are more conve¬
nient to work with. Recall that given a domain U C R2" and k > 0 we set
we
a
into balls
...
kU
Reorganizing the coordinates,
ofR2w.
and
we
{àzgR2"|zg
=
consider the
we
A(au...,an)
=
U(bx,...,bn)
=
abbreviate
An(a)
e
uct
((1
+
e)A(ai,
JO
A(a, ...,a) and On(b)
ellipsoid (1
of Lemma 2.8
(i) by
,an))
e)E(a\,
diffeomorphisms
a,-
4. For
1
(z\,... ,zn)
"
/
/
=
x
•
x
n
^
<
into the
Lagrangian prod¬
the parameter
a
in the
and orientation
area
i
:
i2
a,-
<
then find
e
/
<
1
1
1
D(a{),
i
>
=
g
we
v^^2'
f—'
an
obtain
for all zi
(
•
Replacing
n, we
w
It follows that the restriction of the
a\
(&, ...,b).
D"(l) symplectically embeds
x
F(ai,..., an)
g
a,-
1
e-
=
+ e'
>
!
Rn(x),
R(ai) satisfying
Y^*/(ai(Zi))
f-f
C
into
an).
...,
<
—
n\zi\2
<
\
Un(\),
x
~
i
<
£>(a,)
:
Xi(<*i(Zi))
Figure
Yu=\
a,-,
=
,an) symplectically embeds
...
(i) Define e' by
1
<
0. Then
...
+
J]—
<i<n] cRn(y),
1
(ii) the Lagrangian product A(a\,..., an)
the
|
<xx,...,xn
{0<yi <bi,
=
>
(i) the ellipsoid E(a\,
cf.
Lagrangian splitting Rn(x)x Rn (y)
We set
Lemma 4.4 Assume
Proof,
U}.
ii
1+e.
a,-
symplectic embedding
D(a\)
x
x
D(an)
t-»
R2n
i
<
n,
proof
preserving
4.3.
Embedding ellipsoids
F(ai,..., an) is
to
75
desired.
as
(ii) Define e' by
into balls
X^=i
=
~
e2-
the parameters
Replacing
proof of Lemma 2.8 (ii) by a, and e', 1 D(a, + e') satisfying
Tt\co,(z,)\2
cf.
Figure
4. For
<
+e'
x,
(zi,
for all zt
zn)
...,
s
A(ai,
(x,,yi)
=
,an)
...
±^^^ <±X-^+'i=i
x
co\
A(ai,
...,
D"(l)
x
i+e2
<
1
we
<
in the
and orientation
area
R(a,),
e
i
<
<
n,
then find
(l+e)2.
i=i
It follows that the restriction of the
to
G
and
a
a„)
x
D"(l)
x
con
is
as
In view of Lemma 4.4
:
symplectic embedding
R(a\)
x
x
R(an)
c~>
R2n
desired.
D
may think of
ellipsoid as a Lagrangian product
a simplex
a cube. In the setting of symplectic folding, however, we want
to work with sets which fiber over a symplectic rectangle. We therefore set again
(m, v)
(xi, yi) and define the 2n-dimensional trapezoid Tn(a, b) by
of
we
an
and
=
Tn(a,b)
f (M,i;,x,y) gR2
\
l
=
(u, v)
e
x
R"-1^)
R(a), (x, y)
G
((l
x
-
R"^1(y)
|
f) A»"1^))
x
D"-!(l)
Then
Tn(a, b)
If b
=
a,
we
Corollary
(i)
abbreviate
4.5 Assume
E2n(n,
...
,n,
Tn(a)
e
>
=
=
A(a, b,...,b)x Un(\).
(4.12)
Tn(a,a).
0. Then
a) symplectically embeds
(ii) Tn(a) symplectically embeds
into
B2n(a
into
+
Tn(a
+ e,n +
e),
e).
,n,a) is symplectomorphic to the ellipsoid
Proof, (i) The ellipsoid E2n(n,
F2" (a,n,... ,n), and by Lemma 4.4 (i) and the identity (4.12) this ellipsoid sym¬
...
plectically
embeds into
((1
+
e')A(a,
n,...,n))x
Un(\)
=
Tn(a
+
e'a,
n
+
e'n)
76
4.
for every e'
>
Symplectic folding
in
higher
dimensions
0. The claim thus follows.
(ii) follows from the identity (4.12) and
4.5 the
By Corollary
E2n(n,... ,n,a)
problem
into balls
(ii).
D
of
B2n(A)
embedding trapezoids Tn(n, a)
Lemma 4.4
into
symplectically embedding skinny ellipsoids
equivalent to the problem of symplectically
trapezoids Tn(A). In view of the proof of the
is
first statement in Theorem 3, which will be
completed in Section 5.2, we shall,
however, consider a somewhat different embedding problem. Instead of embed¬
dings of Tn(n, a) we shall study embeddings of the larger set
Sa
R(a)
:=
An~l(n)
x
x
Un-l(\)
c
R2
Rn~l(x)
x
R"_1(y)
3n there exists
a
natural number
x
intoF"(A).
Proposition 4.6 Fix n > 2. For
1(a) and a symplectic embedding
every
cpa:
>
a
Sa
°->-
Tn(l(a)2)
such that the
following
assertions hold.
Ifu
(i)
<
n,
(pa(u,
and
if u
>
a
—
v,x,
y)
n,
(u-a + (1(a)
<pa(u, v,x,y)
,..,
Wa(Sa)\
,.
(u)
lim
-,
-
\)l(a),
v,
^ x, ^ y)
.
.
—r
=
1.
fl-*oo|r»(/(a)2)|
Proof.
The
proof of Proposition 4.6 is more difficult than the proof of the analo¬
gous Proposition 4.3. The reason is that for n > 4 it is impossible to fill a large
(n 1) -simplex with small (n l)-simplices. We shall therefore repeatedly rescale
—
—
the fibers of Sa and fill the cube-factor
n_1(l)
of the fibers of Tn
(l(a)2)
with
the small cube-factors of the rescaled fibers.
Fix
steps
n
we
>
2 and
a
>
3n. We prove
construct for each odd
Proposition 4.6 in six steps. In the first four
number / > 3nn a symplectic embedding \p~i of
the unbounded set
Sœ
:=
{0<m,
0<
v
<
1}
x
An~\n)
x
Dn_1(l)
4.3.
Embedding ellipsoids
into
R2".
5
Step
In
The basic idea behind the
associate to
we
construct
Step
6
into balls
a
>
3n
an
77
embeddings xjfi is explained in Figure 30.
1(a) and use the embeddings i/^/
odd number
symplectic embeddings cpa
verify that these embeddings
Sa
:
Tn(l(a)2)
°-»
we
also meet assertion
(/
Figure
30: The
embedding xfri
-
R2"
1.
Preparations
Fix /
G
2N + 1 with I >3nn. We define subsets
cf.
:=
Figure
{ (m,
Tn(l2) \
(i
31. Define real numbers
ki
v, x,
y)
G
k{
Since /
>
3nn,
we
therefore define
even
ki
—
numbers
{n
=
1)1
Ni
p=
>

3. We may
Ni (I) by
^M
:=
max
|
N + 1
<
Ni-i
:=
max{A^G2N|
7V + 1
<Â:/_i}.
2N
Pi
for /
1 <i </-l.
Nt
g
(i).
(ii).
ki(l) by
±(l-i)l,
:=
find that
-
to
1)/ I1
So
:
Step
Pi
which meet assertion
In
kt
-
,
1
<
i
<
I
-
2,
(4.15)
(4.16)
78
4.
In view of definition
0
(4.15)
di
<
in
higher
dimensions
have
we
1-J---"'+1
:=
and in view of definition
Symplectic folding
(4.14)
we
have
I.
<
kj+i
<
1 <i <l-
k{,\ <i
<
I
2,
(4.17)
2. Then
—
(4.18)
The set
Sqq is symplectomorphic
Fi
via the linear
{0
:=
m, 0
<
Soo
Figure
2.
1}
Aw-1(£i7r)
x
Multiple folding
symplectically
in
embed
a
\^r
ki and Fi, the fibers
contained in the fibers of Fi, cf.
Step
<
(u,v,x,y)
Fi,
->
In view of the definitions of
We
v
x
n"_1(^)
symplectomorphism
a :
are
<
to the set
Figure
(u,v,kix,±y).
A"_1(â:i^)
x
w_1(l/fci)
31: The subset
Pi of
u
—
I
folding
—
Pi
part of Fi into Pi by the multiple folding procedure
following description we shall again neglect the
appearing in the actual construction. We first fold Fi at
right into the y2-direction. The lifting map involved in this
small 5-terms
n
on
the
Fi
F"(/2).
described in Section 4.2. In the
arbitrarily
of
31.
has the form
(u,x2,v,y2)
i->-
[u,x2,v +c(u)x2,y2+
I c(s)ds
F
Jo
4.3.
Embedding ellipsoids
into balls
where the cut off function
next
fold the part of the
This is
y2-direction.
R
c:
image
possible
79
[0, \/(k\n)] has support in [/
->
which y2
on
because I
\/ki
>
at
u
=
2n. The
n
on
-
n,
I].
We
the left into the
length
/
n on the right and at u
n on
alternatingly at u
the left into the y2-direction. We then fold once into the y3-direction. Since Ni is
l—n
even, we do this by folding the part of the image on which y2 > Ni/kiatu
on the right. Going on this way, we altogether fold (Ni +1)"_1
1 times, in which
we fold A7! times into the y„-direction. Denote the multiple folding
embedding
Fi °->- R2" thus obtained by pi. The image of the projection of pi(Fi) onto
Rn-1(y) is contained in the cube Dn~l((Ni + \)/k\), cf. Figure 33. Since N\ is
even, the infinite end of pi(Fi) points into the m-direction. More precisely, the
I
—
>
2n. We fold Ni times
=
of the second floor is
=
—
=
—
last floor of
/î(Fi) is the subset
F[
of
R2
x
Rn~x(x)
Ci
cf.
Figure
x
Rn~x(y)
]0, 1[
x
x
An_1(fci7r)
Figure
x
C\
where
{(y2,...,y„)|f <yj<^,
=
32 and
]n, oo[
:=
j
=
2,...,n],
33.
Define
Si
di-ei.
:=
(4.19)
We choose the 5-terms in the actual construction of the
way that the
M-length
mi
=
By construction,
P2
and fill
are not
as
mi of the
(/
-
n)
the set ßi
much of
P2
as
3.
((Ni
+
Rescaling
I)""1
(Fi ) \ F[ is
-
possible.
to
2^ (/
-
2n)
-
Si.
a
(4.20)
contained in
The fibers
contained in the smallest fiber
need to rescale the fibers of
Step
+
part of Fi mapped
embedding ßi in such
pi (Fi) \ F[ is equal to
Pi. We next want to pass to
A"-1 (kin) x C[ of F[, however,
A"-1 (&27r)
Dn_1 (tt) of P2. We therefore
x
F[.
the fibers
We want to rescale the fibers
An~l(kin)
x
C[
^
<
y}
of
F[
to fibers
A"-1 (£2^)
x
C2
where
C2
cf.
Figure
33.
P2 the fibers
:=
{(yi,
,
y„)
|
1
-
In view of the definition
A"-1(&2:7r)
x
C2
are
<
1,
j
=
2,...,n),
(4.14) of &2 and the definition (4.13) of
contained in the fibers of P2.
We shall first
4.
Figure
32: How
one can
separate the fibers A"-1 (kin)
A"-1(Ä:i7r)
C'(
x
x
Symplectic folding
think about the sets
C[
from themselves
F'v
higher
dimensions
F" and F2.
by lifting
them to the fibers
c;'where
{(y2,...,y„)|^ + ei<y;<^ +
:=
in
and then deform the
separated
Construction of the
lifting Xi
We shall separate
F[
fibers to the fibers
ei,
An~l (k2n)
j
x
=
2,...,n]
C2.
from itself
by lifting its fibers into each y7--direction, j
2,..., n, by l/&i + <?i.
Step 1 of the folding construction described in
Section 2.2 we find a symplectic embedding ßi : ]0, 00[ x ]0, 1 [ ^ ]0, 00[ x ]0, 1 [
which is the identity on {u < n} and the translation (u, v) h> (u + l
n
8i,v)
=
As in
—
on
{m
>
n
inequality
Si), cf. Figure 7. Define si
(4.17) and / > 3nn we find
+
in
n
+
(n
—
\)si
:=
n
+
kindi. In view of the second
l)n (I + kidi)
n
+
(n
<
n
+
(n- l)n(\ +2)
<
3nn
<
I.
—
—
—
4.3.
Embedding ellipsoids
into balls
81
ys
iL
i_
C2
Ni+1
h
*i
k2
Ï2
Ni+1
*i
i_
In view of the definition
(4.19)
of
5i
*i*fe
For
j
=
2,...,
support [n
The
—
therefore find
2)si,
+
n
(j
5i +
a cut
\)si]
—
defined
Gr
+
n
3.
-
ei)-
off function cj
and such that
=
u,
ßi
x
An~l(kin)
x
C[
*-*
R2",
R
:
—>•
[0, l/(&i7r)] with
J0°° cj(s) ds
(u,v,x,y)
=
1/ki
+ ei.
is the
\->
(u', v', x', y')
by
v'
=
v
+
'Y^Cj (u)xj x'j
Xj,
=
,
y'j
=yj+
j=2
identity
on
{u
F'{
cf.
C2 for
have
we
=
and
symplectic embedding
cpi: Im
u'
n we
(j
+
(^jf1), C[, C'{
33: The cubes D""1
Figure
1
Figure
<
:=
n},
{u
maps
>
32. We restrict the
<pi
o
(ßi
x
idn-i)
{u
I, 0
1}
<
<
v
<
to
/
J°
Cj (s) ds,
Pi and translates {u
1}
x
A"-10fci;r)
x
>
j
1}
=
2,...,
to the set
Cj',
symplectic embedding
]0, oo[
x
]0, 1[
x
An_1(Â:i7r)
x
C[
^
R2n
n,
82
4.
to the
Symplectic folding
in
higher
dimensions
intersection of its domain with
identity
ßi(Fi) and extend this restriction by the
symplectic embedding A-i : ßi(Fi) c-^ R2".
to the
In view of the construction of the "translation"
the part of
F[
id2n-2 the M-length
x
of
which X\ embeds into
therefore conclude that the
into Pi is
ßi
equal
Pi is Si. In view of the identity (4.20) we
M-length u\ of the part of Fi which Xi o pi embeds
to
«i
(l-n)
=
((M
+
I)""1
+
-
2) (/
-
2n).
(4.21)
Construction of the deformation cti
The deformation of the fibers
is based
on
the
following
Lemma 4.7 There exists
a:
which restricts
to
the
A""1^)
C" of
x
F{'
to
fibers
A"-1^)
x
C2
lemma.
a
symplectic embedding
]0,Jfc,7T[x]0,^ + ei[
identity on {(x, y) \
y
<
(Ni
-+
+
R2
l)/ki},
restricts to the
affine
map
(x,y)
on
{(x, y) I
y
>
(Ni
^
+
(gx,
l)/ki
+ ei
x'(a(x, y))
for
all
(x,y)
G
]0, kin[
x
|y +
},
1
-
±(Nx
fclfil))
(4.22)
and is such that
y'(a(x, y))
and
< x
+ 2 +
<
1
(4.23)
]0, (Ni + 2)/ki + e\\, cf. Figure 34.
JVl+2
a
X
k2n
Figure
Proof.
Choose
a
smooth function h
34: The map
:
R
—>
a.
R such that
kin
4.3.
Embedding ellipsoids
(i) h(w)
for
1
=
(ii) h'(w)<0
(iii) h(w)
foru;G]^,^±i ei[,
+
forw>»i±l
%
=
In view of the definition
+ ev
of ei and the
(4.18)
We may therefore further
Ch+ei
L
di
<
ei
(lv)
83
^-,
<
w
into balls
have
ki
we
(h(y)x,
T
inequality k2
<
eijt.
<
that
require
1
TJ-^dw
.,
dx.
=
*i
Then the map
]<Uitt[x]0, ^ ei[
+
a:
is
symplectic embedding
a
Denote
]0, oo[x]0, 1[
x
Un-\km)
(m,
where
(x1-, y'-)
«i maps the set
F"
—
=
:=
{m
>
x
y„)
h>-
(u,
v,x'2,
xn, y2,...,
2,
...,
/, 0
<
x'n,
y2,
...,
y'n)
inequalities (4.23),
«i maps the set
(4.22),
onto the set
v
1}
<
x
An~l(k2n)
x
C2,
\jsi
symplectic embedding \jri\ Sœ
embeddings
The
o
In view of the
n.
Pi into Pi, and in view of
...,
R"_1(y),
x
32.
4. Construction of
À/_i
Rn~l(x)
R2
=
iw
Xi (ßi (Fi)) of the symplectic embedding
^»
a(xj, y;), j
j-^—di
required.
nn~l(^ + ei)
...,
n
as
Xi(/xi(Fi)) \ Pi symplectically
Figure
Step
x
v,x2,
Xi(pi(Fi))
F2
cf.
which is
«i the restriction to
by
R2, (x,y)^
-+
ßi_i
o
(ai-2
o
A./_2
o
ßi_2)
R2n
°->-
o
•
•
•
o
(a2
is the
o
X2
o
composition
p2)
o
(«i
o
of
A,i
symplectic
o
pi)
o a.
Symplectic folding
4.
84
Here,
a
and «i
>
higher
dimensions
Step 1, ßi is the map constructed in Step 2, Xi
constructed in Step 3, and the maps ßi, Xi, at, i > 2, are
is the map defined in
are
the maps
constructed in
ai, i
in
2, in
a
similar way
detail,
more
(a,-i
G
ßi,
we assume
embeddings ßj, Xj, a}, j
{ (m, v,x,y)
as
Xi-i
o
o
induction that
by
1,..., i
=
«i. In order to describe the maps ßi,
Xi,
1, where i
—
//,,-_]
o
•
•
ai
o
•
o
<
Xi
we
/
have
already
2, and that the
—
î)(Fi) \
o
u
>
Xi,
constructed
(i
set
-
1)1}
is the set
Fi
{u
:=
(i
>
\)l,
-
0
v
<
1}
<
An~\kin)
x
Q
x
where
({(>2
{te,
,
y«)
|
yn)
|
°
1
-
<?;<£;> j
y,
<
<
yj
l>
;
=
2,...,n]
=
2,
...,
if i is odd,
}
n
if i is
even.
multiple folding map ßi embeds a part of F;- into P,-. We fold Af,- times
il
7r on the right and at u
1)/ + n on the left into
(i
alternatingly at m
the y2-direction, and so on. Since A^;- is even, the last floor of the image of ßi is
The
=
F(
=
—
](i
:=
1)/
-
+ tt,
oo[
]0, 1[
x
x
—
An-1(fc/7r)
C[
x
where
{(y2,...,y„)|
{iyi,
We define
ßi in such
equal
a
yn)
,
8,
di
:=
|
—
way that the
^<yj <^, j 2,...,n}
^ yj 1 f, 7=2,...,«}
1
-
<
<
if i is
-
even.
e, and choose the 5-terms in the actual construction of
M-length
m,-
of the part of F,
mapped
to
ßi(F{) \
F[
is
to
^
(l-n)
=
+
{(Ni
+
I)""1
The maps X, and a, rescale the fibers
An_1(/c,+i7r)
:=
x
Ci+i
{
te,
I {
te,
[
C,+i
ifiisodd,
=
{
•
•
•
•
-
2) (/
An~l(ktn)
-
x
2tt)
-
(4.24)
Si.
C- of the floor
F[
to fibers
where
•
-
•,
yn)
I
yn)
I
1
-
^7
<
'
0
<
yj
yj
<
<
1,
j^,
;
j
=
2,...,
n
}
if / is
=
2,
n
\
if
...,
i
odd,
is even.
Embedding ellipsoids
4.3.
In view of the definition
An~l (kl+in)
The
lifting
direction, j
(4.14) of k,+i and the definition (4.13) of PJ+i the fibers
are
contained in the fibers of
F[
,n,by (— l)l+l(l/k,
2,...
=
embedding X,
{ (m,
v, x,
y)
—
\)l
+
Im ß,
:
F[
g
|
m
F"
+
(n
+
n
in the construction of Ai
as
Pl+i.
by lifting its fibers into each y}
e,). More precisely, we define s, :=
from itself
-
in the construction of Ai that
as
(i
Proceeding
85
map X, separates
and find
+k,nd,
n
Cl+i
x
into balls
R2n
°->-
(i
>
:=
{u
—
>
+
il, 0
<
v
identity
1}
<
Im ß,
on
F[
\
symplectic
a
and translates
to the set
8,}
+
n
il.
<
therefore construct
we can
which is the
1)/
\)s,
—
x
A"_1(^7r)
x
C"
where
{(y2,...,yn)\^
<yj<^ + e„
+ e,
j
=
2,...,n)
if i is odd,
C"
-
{(y2,...,yn)\l-^2-el <yj<\-^-e„
j
=
2,...,n}
if i is
The
identity (4.24)
X,
o
of the part of
M-length
we
u[
The
P, is equal
to
(l-n)
+
=
symplectic embedding
=
Im X,
F(+i
which X, embeds into
P, is 8,. In view of the
u[
of the part of F, which
therefore conclude that the
ß, embeds into
and maps F"
F[
:=
\
{
-
Im X,
:
m
>
il, 0
R2"
2tt).
(4.25)
maps the set Im X, fl P, into P,
<
v
<
1}
same
x
A"_1(^+i7t)
«i if i is
as
way
x
C(+i.
odd, and in
a
similar
even.
Next, the
A^;_i times
on
°->-
-
P, onto the set
The deformation a, is constructed the
way if i is
M-length
((N, + I)""1 2) (/
a,
even.
multiple folding
alternatingly at u
the left into the
last floor of the
F/_j
=
(/
y2-direction,
image
:=
map /x/_i embeds
—
and
1)/
—
n on
so on.
a
part of F)_i into P/_i. We fold
the
Since
right
and at
Af/_i is
u
even
—
(I
-
2)1
+ n,
oo[
x
]0, 1[
x
A"_1(it;_i7r)
2)1
+
n
and / is odd, the
of ßi-i is
](/
—
x
C\_i
86
Symplectic folding
4.
in
higher
dimensions
where
Cl_l
{(y2,...,yn)\l-^<yj<l-^,
=
We define
5/_i
tion of ßi-i in such
to ßi-i
u\. .!
Finally,
(4.16)
F/_,
(Fi-i) \
a
is
(/- n)
=
define si-i
of Ni-i
way that the
equal
we
M-length
ki-in
:=
>
3nn
m/_i of the
part of F/_i mapped
—
-
2) (/
-
(Ni-i + \)n +
2n)
(4.26)
5,_i.
-
In view of the definition
n.
have
h-\
This and /
2,...,n}.
to
((Ni-i + I)""1
+
=
and choose the 5-terms in the actual construc¬
l/fc/_i
:=
j
imply
h-\
—
'
that
(I -2)l
+
n
+
(n- l)si-i
(I
<
-
1)1
-
n.
Proceeding as in the construction of A.,-, i even, we therefore find a symplectic
embedding A./_i : Imyu,/_i c-> R2" which is the identity on Im//,/_] \ F^_x and
translates { (m, v, x,y) e F/_j \ u > (I
2)1 + n + 5;_i } to the set
—
Fi
{u
:=
>
(I
1)1
-
-
v
<
1}
<yj
<
-j^,
n, 0
<
A"_1(£/_i7r)
x
x
Q
where
Q
:=
{
te,
,yn)\0
j
2,...,
=
n
).
M-length of the part of F/_j which A./_i embeds into P/_i is 5/_i + n. In
identity (4.26) we therefore conclude that the M-length u\_x of the part
F;_i which X[-i o ßi_i embeds into P/_i is equal to
The
view of the
of
u\_i
This
fl
completes
=
X1-1
o
(l-n)
=
+
{(Ni-i
the construction of the
ßi-i
o
a/_2
o
A./_2
o
+
\)n~x
-
2) (/
-
2tt)
+
(4.27)
n.
symplectic embedding
pi_2
o
•
•
•
o
ai
o
A-i
o
/xi
o a :
ôo
^^
R ".
4.3.
into balls
Embedding ellipsoids
Step
87
5. Construction of cpa
In view of the construction of the
previous
M-length
Ul
symplectic embedding 0/
four steps and in view of the identities
of the part of Fi embedded into
~
f (l-n)
+
{
+
(/
-
n)
Therefore, the M-length
((N,
((N,
a;
+
l)n~l
+
I)"-1
Xw=l
:=
-
u' °f
2)
2)
me
(I
-
(/
-
R2"
<—>•
in the
(4.21), (4.25) and (4.27), the
P, is equal
-
Sœ
:
to
2n)
2n) +
n
if i
<
I
if i
=
/
-
-
2,
1.
Tn(l2) \ Pi
Part of Fi embedded into
is
/-i
ai
=
J2 (W
Itt +
+
i)""1
2)V
-
2?r)-
~
(4-28)
i=i
Moreover, by construction of 0-/,
0/(M,u,x,y)
=
(m,
fi(u,v,x,y)
=
(u-ai
Using
the definition
v,
3nn,
>
ifu<l
(l-l)l,v,ki-ix,-^y)
+
if
(4.19) of 5j and (4.18), (4.17) and (4.14)
5i
Since /
j-y)
kix,
=
di
therefore find that
we
di
<
ei
—
n
<

ai
n—Si,
-
n.
find that
2n.
<
y
u
—
—
Si. This and the definition
(4.14) ofki and ki-i imply that
fi(u,v,x,y)
=
(u,v,^pLx,Jß=-ly)
fi(u,v,x,y)
=
(m
Before
defining
Lemma 4.8
(ii)
ai
—»•
Proof, (i)
(i)
oo
Fix /
the
as
g
we
set
I
—>
for
f
*;
=
=
=
1)1,
—
we
<pa,
every I
e
v, -x,
further
f y)
u
<
n,
if
u
>
ai
investigate
2N 4- 1 with I
>
(4.29)
—
k,(l),
>
N,
3n7r. As in
=
N,(l),
Step
i
=
1
we
3«7T.
abbreviate
1,...,/-1.
/ + 2 and abbreviate
*,(/'),
N[
=
N,(l'),
i
=
n.
the sequence
oo.
2N + 1 with /
k,
Moreover,
a/+2
(I
ai +
embeddings
<
ai
—
if
!,...,/'-
1.
(4.30)
(a;).
4.
88
in
Symplectic folding
dimensions
higher
By computation,
l~l
the definition
Using
h
i
=
~
=
k~^[
1, ...,/
n^
W
~
~
;
k\,
<N[,
i
read off from definition
so, in view of definition
~
W
~
l'-i-l
=
N't,
(4.15) of Ni and
ki
we
k^'
~
conclude that
\,...,l-2.
=
(4.14)
that
ki-i
(4.31)
^l
=
^V
<
=
/V-i, and
(4.16),
Ni-i
Using equation (4.28)
2
—
therefore find
n^
<
7=7=1
1
1
—
we
2. In view of the definition
—
we
and
(4.14) of ki
Ni
Moreover,
l'~'
>
and the
(4.32)
Ni>-i.
<
inequalities (4.31)
and
(4.32),
we can now
estimate
1-2
ai
/7r +
=
^((M
+
ir-1-2)(/-27T) + ((M_i
l)"-1-2)(/-27r)
+
(=1
1-2
I'n +
<
Y, (Wi +
l)w_1
-
2)(t
-
2tt)
+
((A^_,
I)""1
+
-
2)(/'
-
2tt)
i=\
I'-l
<
l'n +
=
a//.
J2((Nl+l)n~l-2)(l'-2n)
(=1
This proves
(i).
(ii) follows
from
equation (4.28).
In view of Lemma 4.8
1(a)
:=
(ii) the function I
min{/
is well-defined. Lemma 4.8
we
find
{m
<
a
n]
G
2N + 1
(i) shows
that
symplectic embedding ßa
:
and the translation
\-+
(u, v)
:
]n, oo[
| /
R(a)
°->
N,
3nn, a{>a\,
>
/(a;)
—>
=
/. Fix
i?(a/(a))
(u + a/(a)
—
a,
a
>
(4.33)
37T. Since
which is the
v)
on
{u
>
a
<
a/(a),
identity
a
n},
—
on
cf.
4.3.
into balls
Embedding ellipsoids
Figure 7 for the case a
cpa:Sa^T" (1(a)2) by
<
In view of the formulae
assertion
(i)
in
=
fl{a)
and
(4.29)
Proposition
Recall that assertion
(ii)
in
O
(0o
symplectic embedding
id2n-2)-
X
and in view of its
(4.30)
definition,
<pa meets
(ii) in Proposition 4.6
Proposition 4.6
\<Pa(Sa)\
>
\Tn(l(a)2)\
Lemma 4.9 Assertion
define the
finally
4.6.
6. Verification of assertion
Step
We
a/(a).
<Pa
89
(4.34)
is
a
1
as
r-i
of
0
->•
(4.34)
oo.
a -
consequence
\Tn(l2)\MSa,)\
j
claims that
I
as
(4.35)
oo.
-»
|rw(/2)|
Proof.
Since
\Sai\
\Tn(l2)\
Fix
e
G
\fi(Sai)\
1
=
—
=
\Sai\
and
fi(Sai)
\Tn(l2)\-\fi(Sai)\
r—k~\
J
F"(/2),
C
=
1
we
have
\Tn(l2)\MSai)\
—
i
t~\
,.,„
•
(4.36)
\Tn(l2)\
\Tn(l2)\
]0, 1[. The assumption (4.35) and (4.36) imply that there exists Iq
g
2N + 1 such that
\s
1
I
a"
|r«(/2)|
Choosing Iq larger
>
if necessary,
I
>
1 + 2
G
we
tyl-e
for all /
may
G
2N + 1 with /
assume
for all /
G
>
lQ.
(4.37)
>
Z0.
(4.38)
that
2N + 1 with /
a/0. In view of Lemma 4.8 we have a g ]a/, a/+2] for some
/ + 2. Using
2N + 1 with l>k- The definition (4.33) of 1(a) implies /(a)
Assume
/
Vl~e
now
that
a
>
=
90
4.
the estimates
(4.37) and (4.38),
I Ç
Symplectic folding
in
higher
dimensions
therefore estimate
we can
K
I
>
\Tn(l(a)2)\
|F"((/
+
2)2)|
\Tn(l2)\
\Sai\
|r«((/ + 2)2)| |r"(/2)|
\sai\
\Tn(l2)\
i2n
(l + 2)2n
Since
(pa(Sa)
]0, 1 [
e
and
we
\Sa\,
=
we
therefore find
|Sa|
_
|F"(/(a)2)|
Lemma 4.9 follows.
(4.35),
Tn(l2).
Xt(l), 7,(0
fix /
we
D
2N + 1 with /
g
We define the subsets
{(u,v,x,y)ePi(l) \
define subsets
e
-e.
|F"(/(a)2)|
arbitrary,
was
—
_
introduce several subsets of
:=
1
Vl
~
In order to prove assertion
Qi(l)
=
e
—
\<Pa(Sg)\
~
e
Vl
Tn(l(a) ) and $pa(Sa)\
C
>
Since
>
x
and
G
A""1«/
-
=
Pid)\Qid),
Yid)
=
{(m, v,x,y)
lid)
=
{(M,i;,x,y)GÔ!(/)l3'^nn~1(¥i)}'
G
=
1)/)},
of
i
=
3nn and
P,(i) by
l,...,/,
Zi(l) of Pi(l) by
Xi(l)
i
Qi(l)
>
l,...,/,
Qi(l) |
m
£ ](i
$l
-
+
n,il -n[},
»
=
i
=
1, ...,/- 1,
1,...,/-!•
We also set
/
X(l)
=
/-i
\JX,(l),
The sets
Figure
X(l)
33.
and
Y(1)
Y(l)
=
\jYi(l),
i=i
i=i
are
illustrated in
Figure 35,
i-i
Z(l)
\\Zi(l).
=
«=i
and for
Zi(Z)
we
refer to
4.3.
Embedding ellipsoids
into balls
I
n
—
91
(I
n
-
1)1
V
i-\
Figure
35: The subsets
X(l)
=
JJ Xt(l) and Y (I)
i=l
«=i
We recall from the construction of the
Xi(l)
C
]J Yt(l) of Tn(l2).
=
embedding x/ri
that the unfilled space
Pi (I) \ ifi(Sai) is caused by the fact that the size of the fibers of F, is
was needed for folding and contains all stairs, and that
constant, that the space Y (I)
Z, (/) is the union of the space needed to deform the fibers of F[ and the
space caused by the fact that the Ni have to be even integers. Denote the closure
of fi (Sai ) in R2" by fi (Sa,). By construction of fi we have | fi (Sai ) \
| fi (Sa, ) \
the space
=
and
Tn(l2)\JKS^>
C
X(l)UY(l)UZ(l).
We conclude that
Tn(ll) \ fi(Sai)
-n/i2
=
\Tn(lz) \
Lemma 4.10
(i)
\X(l)\<f \Tn(l2)\.
(ii) \Y(l)\
(iii) \z(i)\
<2f\Tn(l2)\.
<
^p|r"(/2)|.
fi(Sai)\
<
|X(/)|
+
|7(/)|
+
\Z(l)\.
(4.39)
Proof, (i)
ln
Symplectic folding
4.
92
—
n
Notice that
Z"_1,
we can
X(l)
C
Tn(l2)\Tn((l-\)l),
cf.
in
Figure
higher
dimensions
35. Since
(/-1)"
>
therefore estimate
\X(l)\
Tn(l2)
<
(l2n
X
=
|r((z-i)Z)|
(l
-
l)»/»)
-
\nl2n~x
<
Tn(l2)
(ii) The definitions
of
F,- (Z) and Qi (I) yield
1^(01
2n
=
10/(01
cf.
Figure
i
—,
l,...,/
=
—
1,
/
35. Therefore,
l-i
\Y(i)\
=
i-i
i-i
JÎF(Z)|
2fY,\Qid)\
=
i=i
2jtZ) 1^(01

then find
±
and so, in view of definition
x
i=l
i=l
(iii) Assume first that
nn2^
<
N, + l
Since i
<
Applying
Z
—
2 and Z
the formula
3n;r
>
(1
we
4tt
k,
(l-i)l
1
r)n
_
have
±
—
i
>
>
4jt
<
1
2tt
^
21
—
—
(n
\)r
—
i
valid for all
r
G
]0, 1[
we can
therefore estimate
n-1
n-l
(t)" M1"*)"
Similarly,
the definition
(4.16)
>l-(n-l)é>l-(n-Df.
shows that
A7;_i + 1
ty-l+i
!
ki-l
>
_
.
_2
ki-\
"
>
ki-i
—
2, whence
(4.40)
4.3.
Embedding ellipsoids
Estimating
as
before,
.n—l
we
(^fef1)""1
0
The definitions of Zi
\n—1
„
-
93
find
/
^
into balls
A)'
>
i
-
(I) and Qi (I) and the
(»
D^
-
estimates
=
i
-
<
_
i
1,...,/
=
v
!«—1(i)|»-i(i)i
10/(01
—
1, and
|Z(Z)|
completes
the
£|z,-(/)i
<
^]T|ß('(Z)|
<
The
(ii)
proof
in
of
27TW
^^
/
—>
1
2itn
<
Tn(l2)
oo,
D
(4.39) and Lemma 4.10
\X(l)\
|F"(Z2)|
\Tn(l2)\
of assertion
1
proof of Lemma 4.10.
\Tn(i2)\fi(saiy
the limit Z
\2^L
l)
yield
i=l
In view of the estimate
Taking
1
now
/-l
i=l
This
{n
c4-41)
so
/-i
=
;
*.•
Df
-
(4.40) and (4.41)
|D"-i(i)\D"-i(£j±i;
lZ.(/)l
(»
we see
Proposition
Proposition
|
|F(Z)1
|F"(Z2)|
that assertion
4.6 is
4.6 is
|
we
find
|Z(Z)|
|F"(Z2)|
~
n
2n
2nn
I
I
I
(4.35) holds true, and
complete.
accomplished.
so
the
proof
94
4.
Symplectic folding in higher
dimensions
Proof of Theorem 3
5
Throughout
this
chapter (M, co)
manifold of finite volume Vol
on
M induced
measure
z i-»- z
this
+
of
a
chapter
(M, co)
R2n by
the
we
again
abbreviate
x
(x2,
=
=
pFa (M,
Proof of lim
As
For
(u,
=
...,
co)
=
again
R2n
measure
denote the translation
we
be denoted
v, x2, y2,
y
=
te,
In
by
xn,
yn)
G
...,
y«)
as
...,
.
R2",
well
as
Rn-\y).
ê
1
We recall from the introduction that for every
is defined
symplectic
ß the
by
before, l^l denotes the Lebesgue
w e
xn) and
(1,...,1)
We denote
symplectomorphism of (R2", coq)
a
coordinates will
ly
5.1
R2".
xw. Of course, rw is
symplectic
connected 2n -dimensional
^ fM con.
=
^ con.
the volume form
(xi,yi,x2,y2, ...,xn,yn)
and
given
a
measurable subset S of
of
w
by
is
a
>
n
the real number
p£ (M,
co)
by
p
pn
Fa
(M, co)
\XP(n,...,n,a)\
=
sup
x
where the supremum is taken
embeds into
plectically
polydisc P2n(n,
...,
n,
over
Vol(M.û))
all those X for which
(M, co). Moreover,
a) is symplectomorphic
we
XP2n(n,
to the set
Rn(a, n).
that the second statement in Theorem 3 of the introduction
Theorem 5.1 For every
e
>
0 there exists
a
a) sym¬
,n,
...
recall from Section 4.2 that the
number ao
can
=
We conclude
be reformulated
ao(e)
>
following property. For every a > ao there exist a number X(a)
symplectic embedding O : X(a)Rn (a, n) <—>- M suchthat
the
n
>
as
having
0 and
a
a
ß(M\<5>a(X(a)Rn(a,n)))
e.
proceed along the following lines. We shall first fill almost all of
finitely many symplectically embedded cubes whose closures are disjoint,
connect these cubes by neighbourhoods of lines. In view of Proposition 4.3
Proof.
M with
and
<
We shall
96
5. Proof of Theorem 3
Figure
we can
and
cf.
Figure
Step
1.
use
Filling
=
the
neighbourhoods
a
thin
polydisc.
embedded thin
symplectically
of the lines to pass from
one
polydiscs,
cube to another,
36.
We denote
C(s)
M with
Filling
then almost fill the cubes with
shall
we
36:
M
by
cubes
the 2n-dimensional open cube
by C(s)
{(xi,yi,
...
Lemma 5.2 For every
,xn,yn)
e
G
R2"
|0
0 there exist
>
<
xt
e
s
<
0
s,
]0, 1 [,
<
an
yt
<
i
s,
=
and
integer k
1,
a
...,«}.
symplectic
embedding
k
\}Ci(s)
y:
^
M
i=l
of a disjoint
union
ofk translates Ci(s) ofC(s)
in
ß(KM\y(\JCi(s))')
Proof.
We
We choose for each
can assume
that the sets
second axiom of
M has
a
Up
countability,
Wi
g
R2n such that the
abbreviate Xi
—
disjoint Darboux
Xpt
°
p
are
and
subcovering.
covering {Vp) of M.
countable
of the open
points
point
'
Ui
M
e.
Darboux chart Xp
a
so
M is
'•
Up
—>
Vp
C M.
Lindelöf, i.e., every open covering of
Since the sets
translates
-*
<
such that
bounded. As every manifold, M satisfies the
We therefore find
T-w, and Vf
charts Xi
G
R2n
—
Ui
Vpi
:=
a
subcovering {V^,}
countable
R2"
UPi
rWl(UPl),
C
i
are
>
bounded,
1,
We have constructed
Vi which
cover
M.
we
find
disjoint. We
countably many
are
pUM,
5.1. Proof of lim
co)
=
1
97
a->oo
We define subsets Vf of M
by
i-i
V{
=
V^V.XU^.
and
Vi
7
=
'
^2-
1
Then
]Jv/
U^
=
«>i
Since the open sets
V,
are
=
m.
*>i
/z-measurable,
the sets V'
are
also
^-measurable.
It
follows that
Since
/x(M)
<
therefore find
oo we
'
\>i
i>i
M such that
m
(5.1)
>
At(M)
/x(M)-^.
>
X>(V/)MV,')
-
(=1
Set
U[
=
x,- (V-/)
smooth, L7' is
and
finitely
£/,, i
C
1,
=
Lebesgue-measurable,
disjoint translates
i
=
1,
...,
V{'
is
^-measurable
and
We therefore find
m.
s
x,_1
e
is
]0, 1[
many
C,,j(s)
of the cube
Since
m.
...,
U[,
C
7
=
1,..., j„
C(s) such that
J'
Y\C,tJ(s)\
z—'
'
>
'
\U;\-—,
'
'
1
=
1
w.
2m
(5.2)
;=i
We set &
=
]C!=i
i«
Since the sets
Moreover, the embeddings Xi
defined
is
'
U't
U[
are
<—>-
M
disjoint, the k cubes Cj,; (s) are disjoint.
are symplectic, and so the embedding y
by
symplectic,
and
V-iXiiCîs)))
=
\C,,j(s)\
and
ß(Vt')
=
|ï/,'|.
5. Proof of Theorem 3
98
In view of the estimates
(5.1) and (5.2)
ß(M\y{jJCl>J(s)^
therefore find
we
=
ß(M)-J2ß(X,(C,,j(s)))
=
ß(M)-J2\Cl,J(s)\
ij
m
ß(M)
=
m
'/
,
J>(v/) + £ (^ \u;\
-
1
=
1
i
=
-
\
J2 KM )
'
l
;=1
m
e
2
e
+ y^
^2m
i=l
e,
=
and
the
so
Let
find s'
e
G
proof of Lemma 5.2
0 be
>
complete.
in Theorem 5.1 and set e'
as
]0, 1 [ and
is
V
M* LJc^')
=
union of k translates
disjoint
^
]0, s'[
G
so
<
C,(8)
and
we
and
C[
we
=
abbreviate
choosing
cubes
:=
s
G
(s'
{z
C,
—
+
=
((s')2n s2")
s)/2.
<
-
For each 5
G
(5.3)
e'.
[0, d]
(5.4)
we
define
((i-l)s-8,-8,...,-8) \zeC(s
C, (0) and
C[
=
]0, s'[ larger if necessary
We define w,
C,(d),i
y!
=
G
R2" through
Yi° *-w,
:
C^
C[
the
c->
^
+
28)},
1,..., k, cf. Figure 37. After
=
we can assume
disjoint.
define the symplectic embedding y[
are
e'.
that
large
k
We abbreviate d
M
C, (s') of C(s') such that
y
s
we
1=1
jk(m\ (]_[£,(/)))
We choose
In view of Lemma 5.2
k
1=1
a
e/3.
symplectic embedding
a
k
of
—
that 2d
identity
M by
M.
<
xWl
s
so
that the
(C,(s'))
=
C[,
5.1. Proof of lim
a—yoo
pUM, co)
We denote the restriction of
Y
=
m
y'
are
=
<
=
C( by
to
y-, and
y'
and
M
we
=
symplectic,
\
write
]J y/: ]J Cj
(
y
we
inequalities (5.3)
=
and
=
1
(y (]J C/))
have ß
(U Q))
M(y(lJCO)
=
99
i' = l
(/ (LI c;)
In view of the
ß(M)
y[
\\Yi\\Ci^
i=l
Since y and
1
=
i'
=
(5.4)
we can
^
M.
l
X! IQ I
z\c'i\Ci\=k {(s')2n
^2"
=
-
and
s2»).
therefore estimate
M(/(LIC;)\y(lJC,-))
+
=
+
A*(Ar\/(lIC;))
(V)2" s2n) +ß(M\y{U Q(s')))
ks2n
+ k
ks2n
+ e' + e'
ks2n
+ 2e'.
-
(5.5)
x,y
Figure
Step
2.
Connecting
Our next
ding
of
37: The cubes
a
goal
C; and C-, i
we
is to extend the
embedding
connected domain. For i
define
—
1, 2.
the cubes
/,and
1, 2, 3, and the lines Li, i
=
straight
lines
Li(t)
=
(
:
/,-
y
:
TJ Q
1,..., k
[ (2i
=
L{(t)
=
-
^
—
1
°^
we
M to
a
symplectic
abbreviate
l)s, 2is ],
R2
x
R""1 (x)
x
(/, 0,0,0)
if i is odd,
j (t,0,0,sl: ,)
if i is even,
Rn~] (y) by
embed¬
100
cf.
5. Proof of Theorem 3
37. Then
Figure
L,(t)
L, (t)
For 5
>
0
N,(8)
we
{I
Proposition
if
t g
[(2i
if
f g
] 2ij
C'l+l
g
define the
,
=
eC[
x
]
-
x
]
—
-
-
"5-neighbourhood" N, (5)
5, 5 [
(]
x
5,5[x(]
5.3 There exist S
G
-
—
5, 5 [
—
8,
d[
'
(5.6)
L, in R2" by
[)n~x
s
}•
+ 8
if i is odd,
[)"_1
if i is
even.
symplectic embedding
U
we
k-\
y',
find
a
^
M.
\y' (JJ C't)
embedding
the set M
smooth
is connected.
Using
k-l
\\X,: \\L,
i=i
\}N,(S)
i=l
construction of the map
(5.6)
5, 5
-
a
of
+
k-1
i=l
this and the inclusions
]
]0, d/S [ and
\}C,(d/3)
p:
x
8, 8[x]s
k
Proof. By
\)s, (2i -l)s
a", 2/j ].
<-+
M\y(]JC,)
i=i
such that
X,(L,(t))
=
yl(L,(t))
ifte[(2i-\)s,(2i-l)s
X,(L,(t))
=
y/+1(L,(0)
if
\ y'
(JJ C, (d/2))
+
d/2],
f[2w-d/2, 2/j],
(5.7)
(5.8)
and such that
X, (L, (t))
G
M
if t£[
(2i
-
l) s + d/2, 2is -d/2]. (5.9)
going to construct a symplectic extension of Xi to a neighbour¬
hood of L i. A symplectic extension of X, to a neighbourhood of L( for i > 2 can
be constructed the same way. We denote by
We
are now
{ei(t),e2(t),...,e2n-i(t),e2n(t)}
the standard
symplectic
=
{£, ^,..., £, ^}
frame of the tangent space
F^^R2".
ppa(M,
5.1. Proof of lim
a—too
Lemma 5.4 There exists
j
1,
—
2n, along the
...,
co)
For
Proof.
and
for 7
=
fj(t)
=
fj(t)
=
[s, 2s]
t G
(5.8) for
j
curve
i
=
Xi(t)
y[
is
2n,
...,
define
1 the assertions
1 and that
forallt[s,2s]
{TL}it)y{)(ej(t))
{TL](t)yï)(ej(t))
we
of symplecticframes {fj (t)},
Xi(Li(t)) such that
—
ftXx(t)
=
1,
=
101
smooth 1 -parameter family
a
fi(t)
and such that for all
1
=
fi(t)
(5.10)
symplectic
we
=
ifte[s,s
d/2],
+
(5.10)
ifte[2s-d/2,2s].
In view of the identities
j-(Xi(t).
and
find
(5.11) for j
=
1
(5.7)
Using (5.10)
f2(t) along Ai (t)
are met.
smooth vector field
a
(5.11)
such that
flit)
and such that
c(t)
:
[s, 2s]
co
->
(TLl(t)y{)
=
(fi(t), f2(t))
a
[0, 1]
_
~
{
[s, s+d/2]
[s, 2s]. Choose
\
°
ü t <2s-
1
if
second smooth vector field
f,,(t,
l2{)
g
e
a
(5.12)
smooth function
such that
C{t)
and define
1 for all t
=
if?
(«2(0)
=
f
0
\
c(t)
2s
-
d/2,
J/3,
f2(t) along Xi (t) by
ift<2s-d/2,
(TLx{t)YC)
We define the smooth vector field
Mt)
t >
=
(e2(t))
ift>2s-
d/2.
/2(0 along Xi(t)by
c(t)fï(t)
+
(1
-
c(t))fi(t).
In view of the formula
(5.12) and the definition of f2(t) the assertions (5.10) and
2 are met, and since
1
(5.11) for j
y2 is symplectic, we find co (fi(t), f2(t))
for all t g [s, 2s]. The subvectorspace Vt of T^^M spanned by fi(t) and /2(0
=
=
is therefore
symplectic subspace. Denote the symplectic complement of Vt in
7\j (f)M by Vf1. Since the linear symplectic group Sp(n 1 ; R) is path-connected,
we find a smooth 1-parameter family of symplectic frames {f?,(t),..., f2n(t)]
a
—
of
j
of
Vf1,
=
t
G
[s, 2s], such that the assertions (5.10) and (5.11)
are
met for all
3,..., 2n. The family of symplectic frames {fi(t), f2(t), fa(t),..., f2n(t)}
Vt © V1
-
TXi(t)M
is
as
desired.
D
5. Proof of Theorem 3
102
Lemma 5.5 There exist ki
oi
>
0 and
Cx(d/3)
:
U
a
embedding
smooth
Ni(ki)
C2(d/3)
U
M
^
such that
(Tl^cti) (ej(t))
and such that
Proof Define
=
embedding
the smooth
a\cx(d/2)
=
a :
ctIl,
W'
all
fj(t)for
Ci (d/2)
=
of
ic
a
>
(5.7)
Ci (d/3)
0 smaller if necessary,
G
[s, 2s]
we
/1 (t)
=
JjÀi(t)
and
U
we can
=
=
=
=
=
=
=
a\c
t
(d/3)
G
=
[5,25].
Similarly,
=
U
C2(d/3)
therefore
Li
ej(t)
X\
if
is
^
,2n.
M
by
y2to the
o
closed set
C2(d/3).
=
i
=
We read off from
1 that the restriction
embedding. After choosing
that c is an embedding.
an
{êj(t)}
we can
...
M
-+
U
assume
j
1,
smooth map
a
of
F^^R2"
by
(5.13)
\,...,2n.
compute
â*/i(0
^*(^K0)
â*(rL](0Ài)(ei(0)
a* (TLi(t)a) (ei(t))
(5.14)
ei(t)
the identities
y2 imply that for all j
êj(t)
Li
U
=
the restriction of
C] (o"/3)
à*fj(t),
â\L
êi(0
for all
to
define the frame
êj(t)
Since
a
=
C2(d/2)
U
(5.8) and the inclusions (5.9) for
and
to the compact set
For each t
Lx
Ci(d/3)VJNi(k)VJC2(d/3)
which extends the restriction of
the identities
U
a\c2(.d/2)
Àl'
Applying the Whitney extension theorem to
Ci(d/3) U Li U C2(d/3) we find ic > 0 and
a:
[5, 2s] ana" j
t G
—
t G
1,
[s,
...,
s
+
(5.10), (5.11)
and
cr\c
(d/3)
—
y[,
2n,
d/3]
U
[2s
-
d/3, 2s].
(5.15)
5.1. Proof of lim
pUM, co)
1
=
103
a->-oo
Define the smooth map
[s, 2s]
a :
x
R2"-1
-*
R2n by
2n
\
J^bMt))
Li(t) +
(2n
Lj(0
=
+J>e,(0-
'
7=2
1=2
In view of the identities
(5.14) and (5.15) the map a restricts
UL,U (C2(d/3) n Ni(ic)). We choose kx
(Ci(d/3) n Ni(ic))
the restriction of
We
can now
a
Ni (ki)
a
smooth
define
embedding
embedding
is
to
whose
an
Ci(d/3)
o-i:
Ni(ki)
U
(5.16)
image
C2(d/3)
U
to
the
>
0
identity
so
on
small that
(k).
is contained in Ni
M
*-+
by
^\Cl(d/3)
For
z
G
Li
we
./
=
^Wk,)
Y\>
then have
cri(z)
off from the
â(a(z))
=
(TLl(t)<ri)(ej(t))
G
[s, 2s] and j
Since the maps
a(z)
=
we
(5.16)
1,
=
...,
2n. The
=
of
=
^2-
M(z). Moreover,
a
that
we
read
(Tiâ) (e}(t))
=
therefore conclude that
(TL](t)a)(TL]{t)a)(ej(t))
=
ai\c2(d/3)
0CC>
and the definition
identity (5.14)
with the definitions (5.13)
(t).
Together
e}
for all t
°
=
(FLi(0<t) (êj(t))
=
proof of Lemma
5.5 is
f}(t)
=
complete.
D
y[
and y2 are symplectic and since the frames {fj(t)} along
the map oi guaranteed by Lemma 5.5 is symplectic on the
symplectic,
Ci(d/3) U Li U C2(d/3). Applying the proof of Lemma 3.14 in [26] to Lx C
Ni(ki) and to the symplectic forms coq and a*co, we therefore find 5i > 0 and a
smooth embedding i/>i : Ni(8i) "^ Ni(ki) such that
X\ (t)
are
set
M(Nl(8)ncl(dß))VLxv(Nl(8)nc2(dß))
Define the
embedding
statement in
pi
:
AT] (5i)
=->
(5.17) and the identities
Ä
I
cri
=
yl
~
by
\c
,„3)
and in view of the second statement in
p*co
=
(o-j
o
=
=
=
y[
ai
o
and
t/>i.
oi\Q
we
«o-
=
=
(d/3)
F
have
fi*(cr*co)
=
COQ,
(5-17)
In view of the first
\
Pl\Ni(8i)nC2(d/3)
(5.17)
Vi)*<w
pi
^
A
an0
î* K«)
and
M
f
P1\Ni(81)nCi(d/3)
id
—
y2'
y2
we
have
104
i.e.,
5. Proof of Theorem 3
pi is
symplectic.
We
pi:
can
therefore define
Ci(d/3)
U
Ni(Si)
a
smooth
C2(d/3)
U
symplectic embedding
M
^
by
pANi{8x)=P^
^lc1(d/3)=W.
Pl\c2(d/3)
(5-18>
Y!'
=
x,y
n
d
Figure
Proceeding
as
d/3
38: The domain
Ci(d/3)
in the construction of pi
/Yi(5i)
U
we
U
find Si
>
C2(d/3)
0 and
of pi.
symplectic
embed¬
dings
Pi:
Ci(d/3)UNi(8i)UCi+i(d/3)
^
M
such that
Pi
where p(-
:
I Q(d/3)
A^ (5()
of the identities
c->
=
Yi
M is
(5.18)
and
a
'
Pi
I Ni (Si)
I C+i (d/3)
(5.19)
i
=
=
(5-19)
Yi+l
1,..., k
—
1. In view
k-1
\\Q(d/3)
U
\\Ni(8i)
«
=
->
M
1
by
P\ci(d/3)UNi(8i)
=
^ddß^NiîY
i
=
t,
.
.
.
,k-
l,
P\ck(dß)
~
Pk~x
\Ck(d/3)
symplectic. In view of the inclusions (5.9) we finally find 5 g
{8i,...,8k,d/S}[ such that the restriction of p to JJ C,(d/3) U TJ Nt(8)
is smooth and
]0, min
Pi
the map
i=l
defined
'
symplectic extension of Xi,
k
p:
Pi
=
5.1. Proof of lim
a-yoo
is
an
Step
embedding.
3.
ppn(M,
The
co)
1
=
105
proof of Proposition
Replacing JJ C,
JJ N, (8) by
U
5.3 is thus
a more
complete.
D
convenient set
previous two steps we are left with filling the set TJ C, U JJ N, (5)
polydiscs. If we would embed a part of a polydisc into the cube Cj by
using the multiple folding technique described in Section 4.2, the x-width of the
last floor of the embedded part of the polydisc would be s, while the x-width of
In view of the
with thin
Ni(8) is
5
<
s.
In order to pass to
fibers of the last floor. This
of
Proposition
replace the
JJ C,
set
U
would therefore have to deform the
we
be done in
can
4.6. For technical
and
C2
similar way
a
shall take
reasons we
JJ N, (5) by
as
set all of whose
a
in
Step
3 of the
proof
different route, however,
a
fibers have x-width 5.
We abbreviate
V
R2" by
We define the subset K of
K
{(xi,yi,
=
and for i
1,
=
K,
...,xn,yn)
k
...,
{z
=
7/
_
'
cf.
_
|0
<
<
x,
8, 0
<
<
y,
K,
are
symplectomorphic to
i(v
v,
"5-halfneighbourhood" H,
f //
f If
v,
i
=
+
in
the cubes
s)],
i
=
=
]0, 5 [
x
(]0,
[
x
]0, 5
x
] 0, 5 [
x
(] 0, 5 [
x
]
5.6 There exists
k
£:
\\K,
1=1
Proof.
We start with
a
C,. We abbreviate
\,...,k-\,
v
-
t)"-1
8, v [)n_1
if i is
\\H,
1=1
k
^
\\C,(d/3)
(=1
k-1
U
\Jn,(8).
1=1
odd,
if i is even,
symplectic embedding
k-1
U
...,n),
l,...,fc.
R2n by
x
5
i
Figure 42.
Proposition
1,
K, of R2" by
define subsets
[(i- i)(y + 5) +
=
and define the
R2"
G
((i-l)(v + s),0, ...,0) \zeK],
+
Notice that the sets
we
s2
ST.
=
106
5. Proof of Theorem 3
a
8
Figure
Lemma 5.7 There exists
a :
which restricts
(x,
y
—
y
+
on
Choose
Proof
(i) h(w)
Since S
<
d/8
]0, 5[
x
]0, v[
Jo
^
]0,
s
+
d/3[
]0, s[
x
to
the translation
(x, y)
h»
—
if
w
and d
G
]0, 5[
< s we
We may therefore further
r
/
d/3
a.
identity on{(x,y)|y<5} and
S }, cf. Figure 39.
{(x, y) \ y > v
U
]v
find
25 +
( ii)
+
symplectic embedding
smooth function h
a
1
=
embedding
the
to
s)
a
39: The
s
require
]0, v[
:
-
[1, (s
—>
+
d/3)/8]
such that
5/3, v[.
by computation
v-28
(s+d/3)/8
<
s
<
that
v.
that
i
—
dw
s.
Hw)
Then the map
a :
]0, S[
is the desired
x
]0, v[
-
Rz,
symplectic embedding.
(x, y)
^
[ h(y)x
çy
,
/
Jo
1
dw
h(w)
D
5.1. Proof of lim
pUM, co)
=
1
a
x
107
a->oo
The
of
(n
l)-fold product
—
into the fibers of
K,
into the fibers of
•
a
x
•
•
in such
embeds the fibers
C, (d/3)
N,(S). We next embed the base of
JJ C,(d/3) U JJ M(5). We denote by X
TJ K, U JJ Ht and TJ C, (d/3) U\JN, (8) onto
We denote
Proof.
t(+
and r~
t(+(m,
i
1,
=
(u, v)
into the base
projections of the
(u, u)-plane, cf. Figure 40.
the
symplectic embedding
the reflection
JJ H,
U
â: X
(v, u), and
\-+
=
(u
+
(i
k. Moreover,
—
\)2s, v),
we
x~(u, v)
define translations
we
=
(u
(i
—
—
\)(v
+
s),v),
abbreviate
lv,
=
](i-\)(v+s),(i-\)(v
//
=
$i-$(v+s)
the map ci :
+
v,i(v
+
+
s)
v\,
+
i
=
l,...,*,
i
=
\,...,k-\.
s)],
properties of the symplectic embedding a guaranteed by Lemma
X —>• R2 defined by
(r;+ oôaoÇor, ) (u,
fo+i or-+l)(u,v)
a(u, v)
a
smooth
symplectic embedding
finally
embedding
We
äxax
define
X
xa:
By construction, £
Proposition
Step
Let
e'
4.
k,
=
5.6 is
e/3
as
to
x
(]0, 5[
U
x
]0,
JJ H,)
y[)"_1
C
^
v
=
JJ K,
C
TJ C, (d/3)
if
u
if
M
If,
Gif,
e
x
U
U
TJ H,
(]0,
s
+
JJ N, (8),
of the
symplectic
d/3[x]0,
s[)n~l
and
proof
so
the
.
of
complete.
U
JJ H,
with thin
polydiscs
d and 5 be the numbers found in the
and
v)
5.7
desired, cf. Figure 40.
be the restriction to
£
(JJ K,
Filling JJ K,
s,
sets
G, cf. Figure 40.
°h>-
In view of the
is
v[)"_1
embedded
by
v)
...,
by £
a
TJ K,
]0,
x
are
and C the
of
Lemma 5.8 There exists
(]0, 5[
way that the fibers of H,
a
s2/8.
For each
a
cpà:Rn(â,n)
>
3n
^
we
previous three steps,
let N(â) and
Rn((N(â)
+
\)n)
and recall that
108
5. Proof of Theorem 3
—
V
Figure
40: The
be the natural number and the
Proposition
4.3
(ii)
V
+
2v +
s
embedding
In view of the definition
we can
(4.11)
we
therefore
have
N(â)
assume
we
have
kâo,
G.
oo as
—
and
we
N(â)
â
1
>
oo.
—>
>
Proposition
ào,
4.3.
By
(5.20)
ks2n'
A^(âo)
>
in
whenever â
Choosing âo larger
>
âo,
and in
if necessary,
that
<
—
°->-
symplectic embedding found
do > 3n such that for all a
(4.8)
N(â)
We set ao
X
there exists
\<pà{Rn(a,n))\
\Rn((N(a) + l)n)\
view of
a:
M
s
8
for all â
>
+ 1
define the function X
:
âo.
[ao, oo[
—>
(5.21)
R
by
s
(5.22)
X(a)
(N (<£)
Proposition
5.9 For each
a
>
ao there exists
+
a
!)*
symplectic embedding
k
ya:X(a)Rn(a,n)
<-+
JJ K,
i=i
k-1
U
]J H,.
i=i
pi (M,
5.1. Proof of lim
a—»-oo
Fix
Proof.
>
a
co)
We set â
ao.
In abuse of notation
X(a).
Moreover,
o(u,
Then
a
(XRn((N
l)n))
+
109
abbreviate N
we
z t->-
symplectomorphism
y)
v, x,
(!«,
=
K, and
=
and
a/k,
=
denote the dilatation
we
define the linear
we
1
=
Xz,
z
R2",
G
and X
also
=
by
X.
R2n by
£x, |y).
jv,
the map
so
of
a
N(â)
=
fà
:
XRn(a, n)
R2"
-»
defined
by
t/t-
i/>â(M>
XRn(â, n)
embeds
symplectically
tions of fa,
and X, and
er
v
=
=
a(u,v,x,y)
fà(u,v,x,y)
=
a(u,
cpô X
Using Proposition 4.3 (i),
picture
in
cally
XRn(â,n)
embeds
(u,
symplectically
the identities
y)
v, x,
Tvly
=
(5.23) and
(â, 7r) into
(5.24) that
=
a(u,
v, x,
y)
fa(u,v,x,y)
=
a(u,
v, x,
y)+
v, x,
(u,
(5.23)
Xn,
m
>
X(â -n),
(5.24)
+
—y) by Çy. Since fa symplecti¬
v, x,
i/>a defined by
°Çy°fà° *Xnly
embeds A.P"
y)
i/>5(«,
i->-
into K, the map
fâ
<
41.
Figure
Denote the reflection
u
(v-jfc%,0,0, (v-jfi^ly)
+
if
cf. the left
the defini¬
find that
we
if
y)
v, x,
o
into K.
sz/S,
y)
v, x,
X
a o
=
K
as
-
£y
well, cf. Figure 41. We read off from
(0,0,0, (v
(v
o
—
j^çî)ly)
j^^, 0,0,0)
if
m
if
u>
<
Xn,
(5.25)
X(â-n),
(5.26)
cf. the
We
right picture
Figure
42. As
Figure
41.
embeddings fa and i/>â of XRn(â,n) into K
symplectic embedding *I>a of XRn(a,n) into TJ i£, U JJ/f/, cf.
in Step 1 of Section 2.2 we find a symplectic embedding
are now
to construct a
in
going
to use the
ß: ]0,v[x]0,8[^]0,v
which restricts to the
(m +
s,
v)
on
Rt
{m
:=
>
{
v
identity
—
(m,
j-^-j-}.
v, x,
{m
on
y)
<
For i
e
v
=
—
1,
+
^zy}
...,
k
XRn(a, n) \ (i
s[x]0,8[
and to the translation
we
—
define
\)Xâ

110
5. Proof of Theorem 3
N+l
Figure
41: The
embeddings fa
and
i/>ô
:
XRn(â,n)
*-+
l)(v + s),
v, x,
K.
Then
XR n(a,n)
]jRi.
=
i=i
We denote the translation
ti. For each odd i G
fa(u,v,x,y)
The estimate
bedding
v, x,
{1,.. .,k
y)
1}
—
(i
—
define the map
we
fa
xfl )
I ti
id2n-2) ofco tf ) (u,
(ß
o
x
(u,
v, x,
:
R{
—>•
y) of R2" by
R2" by
y)
(5.21) and formula (5.26) imply that fa is
Ki U Hi for which
=
42. For each
a(u,v,x,y)
right
i
even
G
y)
The estimate
i Xi
o
(ß
{2,
x
+
(ui,0,0,0)
v, x,
a
y)
smooth
...,
j
k
ifu>X(iâ
P,- is mapped
end of
( Xi oô x~
v,x,
bedding
+
( Xi ofjo
where ui is such that the
fa(u,
(u
\-+
if
m
<
X(iâ
if
m
>
X(iâ
symplectic
—
—
n),
n).
em¬
of Ri into
fa(u, v,x,y)
Figure
(u,
1}
—
(m,
id2n-2)
we
v, x,
o
fa
to the
y)
o
x~
) (u,
(5.21) and formula (5.24) imply that fa is
of Ri into
fa(u,v,x,y)
=
K{
U
H
v, x,
a
y)
smooth
(5.27)
end of
right
define the map fa
j)
—
:
Ri
—>
H, cf.
R2n by
if
m
<
X(iâ
—
if
m
>
X(iâ
—
symplectic
n),
n).
em¬
for which
a(u,v,x,y)
+
(ut,0,0, (v
-
J^p[)ly)
ifu>X(iâ-j)
(5.28)
5.1. Proof of lim
ppa(M,
co)
where m,- is such that the
Figure
42. We
finally
1
=
end of
right
define the
v
Figure
42: The
xi
oô xt
Xi
oi/rô
+
:
)
)
x~
2v +
s
embedding Va
In view of the identities
P( is mapped
the
to
symplectic embedding i/>/t
fa(u, v,x,y)
v
111
°->
y)
if k is
odd,
(u,
v, x,
y)
ifk is
even.
and
^
H, cf.
Kk by
v, x,
XRn(a, n)
(5.23), (5.25), (5.27)
R^
end of
(u,
2v + 2s
s
:
right
3v + 2s
Jjf=1
(5.28)
the
Kt
U
TT*"* Ht.
embedding
k-1
Va:XRn(a,n)
[] K,
^
by
â\R,
is
a
smooth
Step
e
after the
fix
a
>
>
0 be
proof
ao
=
symplectic embedding.
5. End of the
We let
\jHi
i=l
i=l
defined
U
as
The
proof of Theorem
in Theorem
5.1,
i
fi,
set
proof
X(a)
as
in
of
Proposition
5.9 is
complete.
D
5.1
e'
=
e/3
of Lemma 5.2. We choose ao
and define
\,...,k,
=
=
and let k and
ao(e)
as
s
before
be
as
introduced
Proposition 5.9,
(5.22).
Lemma 5.10 We have
\X(a)Rn(a,n)\
>
ks2
e
.
(5.29)
112
5. Proof of Theorem 3
a/k, N
again â
Rn(â, n) °->- Rn((N + \)n) is
Proof.
cpa
:
We set
=
\XRn(a,n)\
and
multiplying
the
kX2n
we
=
N(â)
volume
X2nk\Rn(â,n)\
=
and X
X(a). Since the embedding
=
preserving,
=
kX2n
we
have
\<pâ (Rn(a, n))\
(5.30)
,
inequality (5.20) by
\Rn
((N
+
\)n)\
kX2n ((N
=
\)n)n
+
=
ks2n
find that
kX2n
Lemma 5.10
now
\cpà(Rn(â, n))\
follows from
ks2n
>
(5.31)
identity (5.30)
the
combining
e'.
-
with the estimate
(5.31).
D
Composing the symplectic embeddings tya, £
Proposition
5.6 and
Proposition
<3>a
Using
the estimates
:=
(5.5)
5.3
we
obtain the
and
(5.29)
we
^
M.
find
(<Da (X(a)Rn(a, n)))
ß(M)-\X(a)Rn(a,n)\
(ks2n + 2e'\ (ks2n e'\
ß(M)
=
=
<
-
ß
-
-
3e'
=
e.
—
required
guaranteed by Proposition 5.9,
symplectic embedding
p0£ o*Iv À(a)P"(a,7r)
ß(M\$a(X(a)Rn(a,n)))
This is the
and p
estimate in Theorem 5.1 and
so
the
proof
of Theorem 5.1 is
complete.
5.2
Proof of lim
a—>oo
pf (M,
co)
=
1
We recall from the introduction that for every
is defined
a
>
n
the real number
p^(M,
co)
by
E
p„
FaV
(M, co)'
\XE(n,...,n,a)\
=
sup
x
where the supremum is taken
plectically
embeds into
over
all those X for which
(M, co). Corollary
in Theorem 3 of the introduction is
Vo\(M,cx>)
a
4.5
XE2n(n,
(i) implies
consequence of
...
,n,a) sym¬
that the first statement
5.2. Proof of lim
a-»oo
/rf (M,
Theorem 5.11 For every
co)
following property.
symplectic embedding Oa
e
=
>
1
113
0 there exists
For every
the
a
X(a) Tn(a, n)
:
number ao
a
ao there exist
>
a
=
ao(e)
>
number X(a)
having
n
>
0 and
a
M such that
°->-
ß(M\<3?a(X(a)Tn(a,n)))
e.
<
proof of Theorem 5.1. We
shall first fill almost all of M with finitely many symplectically embedded balls
whose closures are disjoint, and connect these balls by neighbourhoods of lines.
Using Corollary 4.5 (ii) and Proposition 4.6 we can then almost fill the balls with
symplectically embedded parts of X(a) Tn(a, n), and we shall use the neighbour¬
hoods of the lines to pass from one ball to another. The proof of Theorem 5.11
is substantially more difficult than the proof of Theorem 5.1, however. The first
reason is that for n > 3 there is no elementary method of symplectically filling
M with balls. We shall overcome this difficulty by using a result in [25]. The
second reason is that symplectically filling a ball with a part of X(a) Tn(a, n) is
more difficult than symplectically filling a cube with a polydisc. We have over¬
come this difficulty in Corollary 4.5 (ii) and Proposition 4.6. The third reason is
that the fibers of X(a) Tn(a, n) are not constant. This will merely cause technical
Proof.
We shall
proceed along
the
same
lines
in the
as
complications.
Step
1.
Filling
We denote
M
by
balls
the 2n-dimensional open ball
by B(r)
B(r)
Lemma 5.12 For every
e
erty. For each integer k
>
>
=
{zgR2"
0 there exists
ko there exist
I \z\ <r}.
an
r
=
integer ko with the following prop¬
r(k) > 0 and a symplectic embed¬
ding
k
nk:
\}B,(r)
M
^
i=i
of a disjoint
union
ofk
translates
B,(r) of B(r)
in
M(M\%([j5!(r)))
Proof.
s
>
We follow
0 and
a
translates of
Fix
[25, Remark 1.5.G].
symplectic embedding
y
:
JJ
e
>
R2n
<
such that
(5.32)
e.
0. In view of Lemma 5.2
C, (s)
<-*
M of
a
disjoint
we
find
union of
m
C(s) such that
ß(M\y(\JC,(s)^
<
-.
(5.33)
114
5. Proof of Theorem 3
Choose
Zo
G
N
large
so
that for all Z
(Z
Zo,
>
1)»
-
1
>
"
~
We define the
Z
>
integer ko by ko
Zo through the inequalities
(5-34)
T-^7-
"
Z"
3ms2n
=
mnll^.
mn$l- $n
We fix A:
k
<
&o and define the integer
>
mn\ln.
<
(5.35)
The crucial
proved
in
ingredient of the proof of Lemma 5.12 is the following
[25, Corollary 1.5.F]. We abbreviate ç
s/^/n.
result which is
=
Lemma 5.13
(McDuff-Polterovich) There
exist r/
>
0 and
a
symplectic embed¬
ding
n\ln
ßi
:
JJ Bj(n)
7
of a disjoint
union
=
C2n(n)
^
=
D(n)
ofn\
ln translates
Bj(ri) of B(r{)
also
and
=
[30]
Continuing
çBj(ri).
Bij(r)
[34].
=
X-Wl
plectomorphic
morphism
For
with the
For each i
=
{Bj(r))
to
proof
of Lemma 5.12
1,...,
choose u>,-
are
m we
disjoint.
finally
C(s)
—>
Xi. We
z
can now
ß:
x
C2n(n)
denote the dilatation
Ci(s) by
•
x
D(n)
such that
<^3mç2
only
known
proof
h>
-+
C
we
define
R2"
define
a
^
G
r
—
çri and
We therefore find
=
uses
Bj(r)
=
D(n)
a
is sym¬
symplecto¬
(Jn).
R2n
also
JJ C-(j)
i
by
P\b, ,(r)
[25]
such that the mn! ln balls
by ç and
symplectic embedding
çz of
in
<>
]0, */n[.
\\Bij(r)
i,j
the
In view of Lemma 2.5 the disc
]0, +fn[
the square
a:
We
•
2, Lemma 5.13 follows from Lemma 4.4 (i) and Lemma 2.5,
general n, however,
non-elementary, algebro-geometric methods.
see
•
1
C2"(^)\Ä(]JP7(rz))|
Remark 5.14 If n
x
(riocoaoßio ç'1 T|ÜI) |^(r).
o
the translation
5.2. Proof of lim
pi {JA,
In view of the second
of the
co)
=
inequality
family {BhJ (r)].
1
in
115
(5.35)
We define the
we can
choose k members
Bi (r),..., Bk(r)
symplectic embedding
k
m:
\\B,(r)
^
M
i=i
by
r)k
=
y
o
ß.
In order to
verify
the estimate
(5.32)
we
first
use
Lemma 5.13 to
estimate
m
\\Bh](r)\
=
(x'
(?off)(c2"(^))\]J(5-oao/8/)(?,(r/))
J]
i=l
i,j
j
m
=
^^^(^xUßiß^n))
i=i
<
2n
mç
3m ç 2n
e
(5.36)
3"
Moreover, since ß is volume preserving, the inclusion ß
implies
mn\ln\B(r)\
The left
(TJ B,j(r))
C
JJ C,(s)
that
inequality
in
(5.35)
<
and the estimates
|jJ^,;(r)\JJP((r)|
=
ms2n.
(5.37)
(5.37)
and
(5.34)
now
yield
(mn\ln-k)\B(r)\
<
mn$ln -(l-$n)\B(r)\
<
ms2"(l-^)
e
<
(5.38)
-.
Using the fact that ß and y are both volume preserving
(5.33), (5.36) and (5.38) we finally find
and
using
the estimates
ß(M\nk(UB,(r)))
=
ß(M\y(UQ(s)))
e
<
e
e
3+3+3
+
\UCl(s)\ß(UB,,J(r))\
+
\UB,,j(r)\UBl(r)\
116
5. Proof of Theorem 3
The
proof of Lemma
Let
we
e
find
>
an
0 be
as
5.12 is
complete.
in Theorem 5.11 and set e'
e/5.
=
integer ko such that for each integer k
In view of Lemma 5.12
ko there exists
>
rk
0 and
>
a
symplectic embedding
k
\jB,(rk)
Vk-
M
^
i=i
of
disjoint
a
union of k translates
Bt (rk) of B(rk) such that
ß(M\r]k(l[Bl(rk)^
Fix k
balls
to
k^(^nr2kn-(s')2n)
and
(5.39)
bring Proposition 4.6 into play we shall next replace
B,(rk) by trapezoids. We choose s' G ]0, *Jn rk[ so large that
>
ko. In order
e'.
<
we
choose
s
e
]0, s'[
so
large
*
We abbreviate d
(s'
:=
—
à
',
<
the
(5.40)
that
(V)2" s2n)
-
For each 5
s)/(2n).
e'.
<
(5.41)
[0, d]
G
we
define the
trapezoid
7/(5) by
T(8)
Moreover,
we
=
we
2n8)
+
x
Dn(s
define the translates T,
F,(5)
and
An(s
abbreviate
=
T,
{z
+
2n8)
T(d)
=
the linear
AV)xDV)
x
R"(y).
(8) by
Tf
=
T,(d),
i
Figure 44. Notice that T, (8) c T, (8f) whenever 5
larger if necessary we can assume that 2nd <
disjoint.
Composing
R"(x)
C
((i-l>-5,-5,...,-5)UGF(5)},
T,(0) and
=
+
1,
=
<
s
...,
k, cf. Figure 43 and
5'. After choosing
so
that the
s
e
]0, s'[
trapezoids Tf
are
symplectomorphism
-
A"
((5')2)xD"(l)
=
Tn
((5')2)
,
(x,y)»
(s'x, Jy)
p%(M, co)
5.2. Proof of lim
1
=
117
a—too
x
43: The
Figure
trapezoids Fi
T[
and
for
n
symplectic embedding Tn((s')2) <—>- B2n (nrf)
Corollary 4.5 (ii), we obtain a symplectic embedding
with the
a:T(d)
We define the
xWj
(B(rk))
points
We
define the
can now
=
2.
B(rk) guaranteed by
B(rk).
c->
R2n through
u, and wi in
Bi (rk).
=
=
the identities
xVi(Tf)
=
T(d)
symplectic embedding &[ : Tf
and
*-*
M
by
&i
=
Vk
°
We denote the restriction of & to
^
=
LIût: UTi
i=l
i
Since #, #' and r\k
m
=
are
(*' (LI tv)
^
:
Tf
7) by ûi, and
we
rw.
and
M
o
rv.
v'
=
l
M.
<-^
write
\lûi: LITi
i=l
symplectic,
\
o a
*
(117î))
we
=
have ß
(û (J F,))
EIÇ' \ *i|
=
^
M-
i=l
*
=
]T |F(-1
=
fc-^s2",
à (^')2" -s2n)
and
At
(% (JJ Bi(rk))
\ &'
(TJ F/))
=
k
(\B(rk)\ -\T(d)\)
=
k±
(nnr2n (s')2n)
-
5. Proof of Theorem 3
118
In view of the
p(M)
(û (JJ Tf))
ß
=
inequalities (5.39), (5.40) (5.41)
{&' (JJ Tf) \ Û (J Ti))
(nk (TJ Bi(rk)) \ û' (JJ Tf))
1
+ k
<
=
khs2n
+
k±s2n
+ 3e'.
e'
Connecting
We next extend the
+
the
Li(t): [(2i
=
-
e' + e'
(5.42)
trapezoids
G
Tf
ifte[ (2i
Li(t)
G
Tf+l
if
0
JJ F,
\)s, 2is]
Li(t)
>
:
1 ,...,£— 1
=
-
û
^
M to
t->
define
we
R2
symplectic embedding
straight lines
a
Rn_1(x)
x
x
of
a
Rrt_1(y)
(t, 0, 0, 0), cf. Figure 44. Then
Figure
For 5
ß{M\rlk{U Bt(rk)))
(nnr2n (s')2n) +ß{M\rjk(U Bi(rk)))
embedding
connected domain. For i
by Lt(t)
+
k^s2n+k±((s>)2n-s2n)
=
2.
therefore estimate
+ ß
+ ß
Step
we can
we
44: The
define the
Ni(8)
t
e]2is
-
-
\)s, (2i
-
\)s
-
-
2)d [,
and
1)5,
Tf,
2is
,
'
i
=
"5-neighbourhood" Nt(S)
[ (2i
(2n
d, 2is].
trapezoids F(
=
+
]
x
]
"
1,2, and the line Z4.
of
-
Lt in R2n by
8, 8 [
2n-l
K
'
}
p?(M,
5.2. Proof of lim
For later
co)
1
=
119
shall
verify that the left end of N, (8) and the right end of T, (8)
together nicely. In the sequel we denote by pu<v, pua and pv,y the projections
R2n onto R2(u,v),Rn(u, x) and Rn(v,y).
use we
Lemma 5.15 For each 8
G
]0, d] and each i
{(u,v,x,y)eN,(8) \
Proof.
C
We may
{ (M,
X2,
{ (m,
X2,
that i
assume
u
(2i
=
Xn) I
M
=
S
+ 8, -8
...,
x„) I
m
=
s
+ 8,
.
.
-
{1,
...,
l)s
+
k
1}
—
8}
=
{(u
S,
x2
—
8,...,
xn
<
X2,
,xn
...
<
}
8
—8<x2,...,xn and
—
8) \
u
—
8
=
s
m
C
{ (m
—
5,
X2
—
5,
...,
x„
have
F,(5).
C
X2 H
—
we
of
1. We compute that
=
,
.
G
fit
—
8) \
0
<
m
+ 8, 0
+ X2 +
u, x2,
Yxn
...,
<
---
xn
+ X2 +
x2,
<
...,
+ x„
(n
-
xn
1)5 }
and
< s
+ 2n8
< s
+
}
and
---
+ x„
2n8},
i.e.,
Pu,x({
Moreover, ]
-
(u,
8, 8 ["
pv<y({
(u,
v, x,
C
]
y)
-
v, x,
8,
y)
G
Ni (S) I
s
+
G
Ni (S) |
(2n
m
=
s
+ 8
})
(Ti(8)).
C
pu,x
C
/>„,,, (Fi(5)).
1)8 [n, i.e.,
-
m
=
s
+ 8
})
Lemma 5.15 thus follows.
The
following proposition parallels Proposition
Proposition
5.16 There exist 8
G
]0, d[ and
k
p:
a
5.3.
symplectic embedding
k-1
]JT,(8)
i=i
U
\JN,(8)
-+
M.
i=i
Proof. By construction of the map &' the set M \ û'(]_[ Tf) is connected. Using
this and the inclusions (5.43) we find a smooth embedding
k-1
k-1
\JXt: \\L,
i=l
i=i
^
M\tfQj7;)
120
5. Proof of Theorem 3
such that
X,(L,(t))
=
û[(L,(t))
if
X,(L,(t))
=
û'l+1(L,(t))
if t
t G
G
[(2i
\)s, (2i
-
\)s
-
+
(2n
-
l)d/2],
[2is -d/2, 2is],
and such that
X, (L, (t))
For i
G
=
M
1,
\
û'( JJ T, (d/2))
...,
k
1 and 5
—
f,(8)
cf.
Figure
45.
Replacing
=
Fi(d/3), Fi (d/2),
we
find 5
>
0 and
a
[ (2i
we
-
l)s
+
(2n
5.3
l)d/2,
-
define the truncated
{z£T,(8) \u <(2i-1)5
...,
proof of Proposition
t G
[0, d]
the maps
Ci(d/3), Ci(d/2),
in the
G
if
y[,..., yf
+
2is
-
d/2 ].
trapezoid
5},
and the sets
Ck-i(d/3), Ck-i(d/2), Ck(d/3), Ck(d/2)
by
the maps
û[, ..,û'k
.
and the sets
ÏVl(d/3), fk-i(d/2),
...,
Tk(d/3), Tk(d/2)
symplectic embedding
k-i
p:
]j(F!(d/3)UATi(5))
U
Tk(d/3)
^
M,
i=i
cf.
Figure
45. We define 5
by
5
•
=
mm
h
d/3 1
>.
{ 8,
2n
1 J
I
-
Then 5
<
d and T,
(8)
C
T, (d/3) for all i
=
1,
...,
the domain
k
k-1
\}T,(S)U \JN,(S)
1=1
is
as
desired.
1=1
k
—
1. The restriction of p to
5.2. Proof of lim
a-^-oo
pf(M,
co)
1
=
121
x,y
n
-d/3-d
Figure
Step
45: The set
Fi(d/3)
U
Ni(8i).
3. The choice of k and of ao
We recall that e'
=
Lemma 5.12. We
now
e/5,
and
we
choose the
choose the
integer
integer ko
k such that k
(**)'-»
>
as
after the
2 and k
>
proof
of
ko and
1
(5.44)
ß(M)
proof of Lemma 5.19 below. Let
and d be the numbers associated with k after the proof of Lemma 5.12, and let
be as in Proposition 5.16. For each â > 3n we let 1(a) and
The
inequality (5.44)
will be crucial for the
<Pâ-
be the natural number and the
,B
Sä
s
5
(l(à)2)
symplectic embedding
found in
Proposition
4.6.
We abbreviate
e'
1
By Proposition
4.6
(ii) there
exists
âo
>
n\
(5.45)
k s2n'
37T such that for all â
>
âo,
\<Pâ(Sâ)\
|F«(Z(a)2)|
In view of Lemma 4.8 and the definition
ever
â
>
âo
and that
1(a)
-»
oo as
â
—>
>
(4.33)
oo.
(5.46)
q.
we
have that
1(a)
Choosing âo larger
>
Z(âo)
when¬
if necessary,
we
122
can
5. Proof of Theorem 3
therefore
that
assume
l(â)
We set ao
=
knâo.
In the
>
for all â
n-
>
5
sequel
fix
we
>
a
â0.
ao. We set
(5.47)
â
=
f^.
k
]J S,
4. The set
Step
and the choice of À (a)
i=i
R^
With each & -tuple
(mi
by
h u,. We say that
v,
=
mi +
for all i and if i>£
=
,
uk)
...,
g
a
a
as
«2,
follows.
•
•
•,
G
Assume that i
"i-i- We set
y'o
=
Vj
=
max{ j \v}
—>
/(mi)
=
oo as
Mi
...,
...,
—>
-
The function
/
/(Ml)
Ujfc
strictly increasing.
a we
that
1,
=
(l "-ff''
=
=
:
we
...,
i
admissible and meets the identities
(5.48)
wi numbers u2,
have
—
.,uk
..
constructed
already
1. We define u,
by
UX
]0, a[
Since
—
/(mi)
conclude that there exists
the k -tuple
such that
i=2,...,k.
"1'
\-Uj, j
mi +
By construction,
a.
=
{2,... ,k] and
G
a).
<
is then continuous and
/(mi)
,
]0,a[. We inductively associate with
M,
where
A;-tuple (mi
vk) defined
&-tuple (vi,
uk) is admissible if u, > 0
unique admissible k-tuple (ui,..., uk)
M'(1-iTip1
Fix M]
associate the
a.
Lemma 5.17 There exists
Proof.
we
a
R defined
—
0
unique
by
as
mi
mi
g
—>
0 and
]0, a[ with
associated with this mi is
,uk)
(ui,
(5.48). The proof of Lemma 5.17
...
is
complete.
D
Let
(mi,
...,
abbreviate v,
=
R2 xR"-'W
uk) be the a:-tuple guaranteed by Lemma 5.17. As before we
0. We define subsets S, of
+ u,, and we set vq
ui +
•
=
xRn-!(y)by
l[xA"-1(7r)xD"-1(l),
Si
=
]0,i;i[x]0,
S,
=
[vi,«1[x]0,l[xA"-1(3r-^_1)xG"-1(l),
2
<
i
<
k.
pi (M,
5.2. Proof of lim
Then
Tn(a, n)
C
JJ S(,
cf.
1
=
123
46. Notice that
Figure
r«-1
IÇ.I
\î\
The identities
co)
-
—
„.
,n-l
J£—(l
^
(„_i)i
"«
1
B=LY
-
^
a
i
<
k.
<
(5.48) therefore imply that
\Sx\
=
46: The set
Figure
(5.49)
\Sk\.
=
---
JJ 5,-
for k
=
4.
Lemma 5.18 We have
ao
<
<
j^
<
«l
<
"2
<
Prao/ The first inequality in (5.50) is equivalent
inclusion
Tn(a,n)
an
JJ Si
C
^
<
mi. The
(5.48). Finally,
(5.50)
W
assumption
(5.49) yield
to our
\Tn(a,n)\
inequalities
the identities
and
so
We
uk
<
now
< a.
The
TjSf
<
ui
<
U2
ikISil
=
<
•
•
<
k
(n
-
i>k)r
=
(n-\)\
k\Sk\
=
JJ Si
C
Sa yield
\USi\
-f?=.
n
<
1)!
uk follow from the identities
and the inclusion
(5.49)
=
uin
vn-l
kuk
ao
.,._«—1
=
so
^
n—l
n\
and
and the identities
a*
\Sa\
n-l
=a
(n
-
1)!
D
define the real number X
=
X(a) by
\XSi\
=
\Ti\
q.
(5.51)
124
5. Proof of Theorem 3
Lemma 5.19
x\JSi\XTn(a,n)
i
Proof
=
'
<
l
In view of Lemma 5.18
JJ St
have mi
we
(a
Tn
C
<
<
u2
+ uk,
n
+
<
ûk)
uk, and so
(5.52)
,
i=i
cf.
47.
Figure
n
+
ûk
»
Figure
47:
Tn(a,n)
JJ St
C
Tn
C
(a
+ uk,
n
+
u
f uk).
i=i
Since the
embedding
&
:
ß(M)
In view of the inclusion
and
(5.53)
we can now
\J T,Ie-»
>
M is
symplectic,
fi(û(UTi))
(5.52),
the last
=
£|7}|
inequality
in
we
=
have
k±s2n.
(5.53)
(5.50) and the estimates (5.44)
estimate
\USi\Tn(a,n)\
<
\Tn(a
=
*£ ((a + uk) (1 + *)""l
=
^(0+^r-o«
<
n
^-
-r^n
+ uk,n +
—,—tttt e a
—
l
1
j
„
fuk)\Tn(a,n)\
-
a)
5.2. Proof of lim
p?(M, co)
1
=
125
a->-oo
Using
the definitions
(5.51) and (5.45) and the second inequality
in
(5.50)
we
find
that
I FJ
X2n
n
=
——
I Til
<
\Si\
?2n
—
»!
k?2n
<
_
=
—-
Kn~x
Ul(n-1)\
\Si\
nuinn~l
„.
ann-v
~
We conclude that
\X\jSi\XTn(a,n)\
and
so
Step
the
5.
of Lemma 5.19 is
proof
Embedding
Proposition
X
TJ 5,-
a
U
JJ N, (8)
k
k-1
Va:X\Js,^ \}Ti(8)
i=l
We start with
the sets
XSi by
more
e1
<
symplectic embedding
k
Proof
n)\
complete.
JJ Tt (8)
into
5.20 There exists
X2n | J St \ Tn(a,
=
U
i=l
introducing
several
JJ^(5).
i=l
geometric quantities
and with
replacing
convenient sets.
5.A. Preliminaries
For i
=
1,..., k
we
set
hi
Notice that
We define a,
=
hi
h2
>
mi,
l(ai)
âo
finally
—
hk.
^
the estimate
define À,
we
have ai
(5.47)
>
m(-
>
Proposition
>
mi
4.6
âo.
We
by
Kai).
=
shows that
/,•
We
>
associated with a, in
h
>
...
and in view of Lemma 5.18
denote the natural number
Since a,-
>
by
a-
=
a
hin is the x-width of Si and that
1
Then ai
"i-i
\
=
1
>
7rf.
(5.54)
fr
(5.55)
by
Xi
=
126
5. Proof of Theorem 3
Lemma5.21
X,
Since
Proof.
>
\T,\
=
«Jh,X
^-
and
>
âo,
>
\Tn (if)|
and
|F, |
and since a,
8
the estimate
-s/hfnX.
=
+y
A2"|f"(z2)
=
(5.46)
shows that
K
|F»(Z2)|
Together
with the definition
have
we
>
q.
(5.51) of X and the
identities
(5.49)
|A.5i|
X2n\S,\
we can now
esti¬
mate
|Sa,|
\T"(lf)\
hfn\S,\
=
~
Xf2n\T,\
and
so
>
q
=
=
~
~
\Ti\
'
\T,\
the first statement in Lemma 5.21 follows. The second statement follows
from the first one, from the definition of X, and from the estimate
We define the linear
symplectomorphism ß
ß(u, v)
Moreover,
we
set m,
j 2
=
5,
—
u, and v,
=
=
üi
R2 by
of
ßu, {v).
+
+
(5.56)
ü,,
(ßxid2n-2)(XSl),
:=
(5.54).
and
we
abbreviate
\<i<k.
Then
Si
=
]0, ûi[x]0,5[x
S,
=
[0,-1,0,[
]0,s[x
x
We shall next embed
A"_1(7TÀ)
x
ü""1^),
/\n-\h,nX)
x
Un-l(X),
2

X.
s
c->
=
Fi
Xili, and
we
read off from the first statement in
Therefore,
ti
:=
-y-ai
—
mi
>
0,
pf;(M, co)
5.2. Proof of lim
1
=
127
a-^-oo
and
so
the affine
ßi(u, v)
embeds
pu,v(Si)
of
symplectomorphism ßi
into
]0, Xiai[
x
R2
\li(u
=
+
defined
±v)
ti),
]0, Xi[. Using
by
that
once more
Ai
>
X
we con¬
clude that
(A.]"1 oG8i id2„_2)) (5i)
C
x
We define the linear
ox(u,v,x,y)
Composing the affine embedding
plectic embedding
{j-u,hv, j-xx,hy).
=
À^1
obtain the
:=
We abbreviate si
we
o\
s
=
Xi
o
—
o
cpax
o
Xf1
S{
=
(2i
-*
diffeomorphism
7i
o
(ßi
x
id2n-2) '-Si
(u-vi+si,v,^x,ny)
=
embedding fa
1,
with the sym¬
Sai
^
Fi.
Xi. Using Proposition 4.6 (i) and the definitions of ßi,
ai
:
JJ S,
-+
...,
—
k
l)s
we
—
if
u>
]J F, (5) U JJ TV, (5),
i
=
vi-^-n.
7-1
j
i=l
=
°->-
Tn(l2)
Tn(l\)
7
For i
Si
find that
fa(u,v,x,y)
5.C. The
'•
symplectic embedding
fa
andÀi
<-»
id2n-2)
x
4.6 and with the linear
0-10A.1:
we
(ßi
o
<pai:Sai
guaranteed by Proposition
R2" by
of
symplectomorphism oi
Sai.
l
i
=
j =2,...,k
l
abbreviate
X{,
d{
=
min
I
y,
5
J,
Proceeding by induction we fix j g {1,...,
already constructed a symplectic embedding
7
j
£,•
k
=
—
min
1}
{m,-
and
(=1
d,-,
f-n).
assume
that
(5.57)
we
have
7-1
fiMî ^WTiwuLJNi(5)
i=l
—
i'=l
(5-58)
5. Proof of Theorem 3
128
which is such that Im
fj(u,v,x,y)
i/>7
C
{(u,
y) |
v, x,
Sj)
;
which is such that
fa+i(u,v,x,y)
=
(u
-
i
i/>; by
formula
{(u,v,x,y)
Vj+i +Sj+i,
(5.59)
Sj,
\
u
<
s/+i}
J.— ±x,
v,
to the smooth
(5.60)
l
=
if
We extend
—
{J ^ (8) U ]J Ni (8)
^
C
Vj
7
i=l
Imi/>;+i
>
M
extension
symplectic
7+1
i=l
of
a
(5.59)
and
y/hj+inyj
>
M
(5.61)
Vj+l -Sj+i.
symplectic embedding
7 +1
fa:
\JSi
i
and
we
denote
Fj
cf.
=
Figure
by F'-
=
R2n,
^
l
the end
\j/j(Sj+i)
=
{(m,
v, x,
y)
G
Im
fa \
sj

=
O+i
Step 1 of the folding
plectic embedding
As in
ßj
:
[sj,sj
+
:=
=
Xj+ilj+i,
^Jhj+iX.
t/t^'+i
~~
and
we
read off from the first
Therefore,
"7+1
>
(5-62)
°-
construction described in Section 2.2
üj+i[
x
]0, s[
<-+
ysj, 2js + iß±aj+iy
x
we
find
]0, s[
a
sym¬
5.2. Proof of lim
pf (M,
co)
1
=
129
Tj(8)
Sj (2j
-
Tj+i(8)
—JNj(8)
l)s
Figure
which restricts to the
identity
on
2js
48: The end F'-.
| (u, v)
\ Sj
-
+
2js
—
Sj + tj+i, v)
{(u, v) \
on
Sj +
>
u
dj+i},
(5.63)
and is such that
Imßj
n
{(2j
Imßj
n
[2JS-8
-
+ 8
\)s
+
<
^
<
m
Imßjn{2js-^ + 2^
cf.
Figure
<
u
<
8}
C
{0
f}
C
{s
<2js}
C
{0
2js
-
2js-
<u
<
v
-8
<
<
v
8),
<
v
(5.64)
<s},
(5.65)
8},
<
(5.66)
49.
Lemma 5.22 We have
(ßj
Proof.
x
id2n-2)
(Fj)
n
{
(m,
v, x,
In view of the inclusion
Pu.v((ßj
x
id2n-2)
(Fj)
n
y)\Sj
(5.64)
{ Sj
we
< u
<
< m
<
2js
-
5
x
C
Tj(S)
U
Nj(8).
have
2js
-
8
})
C
pu.v
Moreover, the formula (5.59) implies that the fibers of F'.
(hj+i-^=±XAn-l(n) J
}
jh]nXnn-l(l)
=
A"-1
are
(wj)
(Tj(8)
equal
x
U
Nj(8))
.
to
Dn~l
L/hfnX)
130
5. Proof of Theorem 3
dj+l
Sj
(2j
—
l)s
+ 8
2js
—
8
where
we
In view of the
are
ßj.
abbreviated
ij+i
Wj
fibers
tj+i
2js
2js
49: The map
Figure
+
inequality hj+i
=
and the second estimate in Lemma 5.21 these
hj
<
(5.67)
X.
contained in the fibers of
Nj(8).
Lemma 5.22 thus follows in view of
Lemma 5.15.
5.C.2.
Rescaling
We define Vj
the fibers
by
VJ
In view of the
=
inequalities (5.54)
(O+i
we
ej
J
-
have
=
D^-
Zy+i
>
—JhjnX
v :
V
7
3, and
(5.68)
so
Vj
>
2. We abbreviate
(5.69)
pjf (M,
5.2. Proof of lim
and
co)
=
1
0
<
131
define
we
JhfnX,
C
{
C"
yw) I -y/hjnX
y,{(y2,
2ej
-2ej,
2,
yn) I ~2ej
y,- <-ej, i
{(yi,
{(yi, ...,yn)\0<yi<ej, i 2,..., },
(J2,
•
•
•,
yn) I
y,-
<
C"
c
cf.
•
—
<
-
•
C7
i
•,
=
Figure
},
n
i
...,
n
=
,n],
2,
},
n
=
7 +1
...,
<
<
,
2,
50.
A
hjnXC;
y2
y/hjnX
ei
Figure
Our next
{m
>
2js
in
Step
at
m
=
fibers
fibers
goal
—
5}
—
Cj, C'f,
to
the fibers
proof
8 +
of
A"-1
(vjwf)
Proposition 4.6
j^ to the fibers
An~l(vjWj)
An~l(VjWj)
x
C"',
x
Cj+i.
and
x
Cj+i.
x
C7+i
for
C- of
(ßj
We shall
use
n
x
=
x
C"-,
3.
id2n-2){F'\
the
and first lower the fibers
A"-1 (wj)
finally
C" and
1(wj)
is to rescale the fibers An
3 of the
2js
50: The cubes
same
over
method
A"-1 (wj)
x
as
C-
then deform these fibers to the
lift these fibers at
u
=
2js
-
|
+
^
to the
5. Proof of Theorem 3
132
The
lowering
Using
map cp
the definition of e} and the
2e}
using
S
d
<
2ejVjWj
Combining the
2hJ+inX2
=
estimates
2,
=
support
2
estimate
we
(5.70)
2e}Vj,
<
\_2js
we
n
...,
8 +
—
+
1)^-,
—
2js
8 + i
—
82
<
<
j^s.
(5.71)
±s.
off function c(
cut
a
<
find that
we
y/hfnX)wJ
therefore find
(i
{^fhfnX)
<
(5.70) and (5.71)
(2e}
For i
jh}nX
+
>
the definitions of e} and w}, the second statement in Lemma 5.21 and
^ we estimate
and
<
inequality v}
j^ ]
R
:
0,
—
^-
with
and such that
/OO
/
cf(t)dt
Jo
The
=
2e}
+
y/hfnX.
symplectic embedding
<pj :1mßj
defined
An~l
x
(wj) xC'j
R2",
^
(u,
v, x,
=
yl-
I
h*
(u', v', x', y')
by
n
u'
y)
v'
u,
=
v
-
^2 c7d<-)xi,
maps the fibers A"
An-l(w,)
x
J
!
(w})
x
C'
pu
x[
=
over
y[
x,,
=
the base
{(u,
v) \
cf(t) dt,
u
>
2js
—
=2,
i
|}
...,
n,
to the fibers
C".
J
The deformation a}
The deformation of the fibers ,Kn-l(„,
\
An~l(wj)
based
on
the
following
x
r"
C"
tn
to
tho «Köre
the
fibers
Ai-ir,,
a
symplectic embedding
]0, Wj[
x
]
-
2ej
-
^/hfnX, oo[
^
,.,
\
An-l(VjWj)
lemma,
Lemma 5.23 There exists
a::
„
R2
^
x
r'»
C"
is
p%(M, co)
5.2. Proof of lim
a-»oo
which restricts
on
{(x,
y) |
<
y
the
to
identity
on
(x, y)
h>
2ej},
—
G
]0, Wj[
133
{(x, y) |
>
y
0},
restricts to the
affine
(vjx, ±y + j:2ej ej)
map
(5.72)
-
and is such that
x'(a(x, y))
for all (x, y)
1
=
<
]—2ej
x
y'(a(x, y))
and
Vjx
^fhfnX, oo[,
—
>
—
(5.73)
8
c/ Figure 51.
y'
.
,.J
J
y
,
^-
BjfPPPM^/fejTr
A,
Figure
Proof.
Choose
a
smooth function
(i) f(w)
=
Vj for
w
(ii) /(u;)
=
1
w >
Since v/
>
2
we
for
<
/
51: The map
R
:
[1, v/] such that
—>-
0.
have
We may therefore further
/
a.
-2ej,
2e7^y
(iii)
-^1—7—X
-£;-^^|
a
7T^
Ju;
require
<
ej
<
that
e^.
=
Using (i) and (iii), the definition (5.69) of ej,
inequality in Lemma 5.21 we can estimate
/ 2ej-^/h]nk
2ej-
1
f(w)
the
inequality Vj
>
2 and the second
1
dw
=
—JhjnX
Vj
+ ej
=
—JhjnX
vi
<
134
5. Proof of Theorem 3
We conclude that the map
]0, Wj[x\-2e j-^/hfnX,
a:
is
symplectic embedding
a
In view of
(5.72)
the
:
a}
defined
ooy^R2,
which is
as
If(y)x,
h>
/
——dw
required.
J
D
symplectic embedding
(cp-
(ßj
o
(f;)
id2n^2) )
x
R2n
^
by
aj(u,v,x2,y2, ...,xn,yn)
maps the fibers
An~l(Wj)
An-x(VjWj)
Cf.
The
(x, y)
lifting
x
x
C"
=
over
(u,
a(x2, y2),
v,
{(m,
the base
v) |
>
m
a(xn, y„))
...,
2js
—
|}
to the
fibers
cpf
In view of the estimate
(5.71 )
[o, ^j]
[2js
with support
we
|
-
find for i
+
(i
-
=
2,
1)£,
n a
...,
2js
§
-
cut off function
i£ ]
+
cf'
:
R
—>
and such that
/•OO
/
cf~(t)dt
Jo
=
2e}.
'o
The
symplectic embedding
cpf: (aj
defined
u'
=
u,
ocpj
o
(ß}
x
id2n-2))(F'])
^
,2n
R2n,
(u,
v, x,
y)
\->
(u', v', x', y')
by
v'
=
v
+
Y] cf~(u)x, x[
=
,
x,,
y[
=
y,+
~
i=2
An~1(vJWj)
An-\v]W])xCJ+i.
maps the fibers
We abbreviate the
cpj
=
x
C"
over
the base
I
Jo
cf~(t) dt,
{(m, v) |
m
>
2js}
composition
cpf
occjo cp~
o
(ßj
x
idln-2)
:
F'}
^
i
R2n.
=
to
2,
...,
n,
the fibers
pf (M,
5.2. Proof of lim
co)
=
1
135
Lemma 5.24 We have
(Fj) n{(u,v,x,y)\Sj <u<2js}
cpj
We fix
Proof.
(u,
v, x,
y)
(ßj
e
(u', v', x', y')
Then u'
=
id2n-2) (F'f)
x
(cpf
=
o
n
{sj
U
Nj(8)
<
< u
<pj)(u,
o
aj
F;(5)
C
v, x,
U
2js},
Tj+i(8).
and
we set
y).
u.
Assume first that
<
m
2js
—
Then
8.
(u', v', x', y')
=
(u,
v,x,
and
y),
so
Lemma 5.22 shows that
(u',v',x',y')
Assume
and
now
that
2js
—
8
<
<
2js.
Tj(8)UNj(S).
We read off from the inclusions
(5.66) and from the definitions of <p~, aj
v'
cf.
u
G
G
cpf
and
(5.65)
that
]-5,s + 5[,
52.
Figure
v
5
+ 5-s
n
Figure
Moreover,
that x'
g
2js
-
52: The set pUtV
we
have
x
An~x(VjWj).
G
8
(<f>} (F'f) fl {2js
An~x(wj),
Using
and
so
-
8
<
the first
the definitions
u
<
2js
})
inequality
for
in
n
=
3.
(5.73) implies
(5.68) and (5.67) of Vj and wj, the
(5.55) of Xj+i we compute
first estimate in Lemma 5.21 and the definition
»M
=
dj+i
~
l)y T^^x
=
dj+i
~
V)y/hj+~\l
<
h+l^j+l
=
s,
136
5. Proof of Theorem 3
and
so
x'
Finally,
we
have y
and the second
C
e
Dn_1
=
inequality
in
G
A"-1^).
(jhfnX),
find
we
y
the definitions of
so
y'
G
]
yfhfnX
<
8
that
(5.73) imply
second estimate in Lemma 5.21
and
^fhjnX[n~l.
5,
—
<
and
s,
and
cpf
Using
the
cpf
so
n-l
g
]-8,s[
We conclude that
(u', v', x', y')
The
G
]2js-8, 2js]x]-8,s+8[x
proof of Lemma
Recall that
ßj
5.24 is thus
is the
identity
tity
~
on
\(u,v)
symplectic embedding cpj
symplectic embedding
/;t+t~\
~

can
R2" by
the iden¬
ITT % \,
fj
cpj
on
^ (5;+i).
(5.68) of Vj
we
-
1)-J=I*,
find
jj^^hjfiny^j
m
>
Vj
+dj+i.
tyj+i
the set
]2js, 2js fâaj+i[
+
In view of the construction of
<t>j
there¬
=
vj + 2js + tj+i,
by Fj+i
F;+i(5).
C
We
on
the definition
(5.59) and (5.63) and
-^ \.
id
if
We denote
jf"1
complete.
fore extend the
to the
An~\s)
[Fjj
n
cpj
we
x
]0, s[
x
An-\vjWj)
have that
{(w, v,x,y)\u> 2js]
C
Fj+i
x
CJ+i.
(5.74)
5.2. Proof of lim
a-±oo
2js, v, x,
of R2" by
co)
1
=
137
cpj (F'-) is equal to the right face of F/+i. We are going
FJ+i into Tj+i. We denote by Xj the translation (u, v, x, y) i->- (u +
y). Moreover, we define the linear symplectomorphisms <r;+i and £7-+i
and that the
to embed
p5(M,
right
face of
aj+i(u,v,x,y)
(jû, lj+iv, j^x, Ij+iyf
=
and
Çj+i(u,v,x,y)
Using
the definitions
the first
inequality
(u,v,(lj+l*l)h+lx, ^+l~J)h+ly).
=
(5.68), (5.67), (5.69) and (5.55) of Vj, Wj, ej and Xj+i and
in Lemma 5.21
we
find that
(^ôA-ôôr-^Fy+i)
]0, aj+d
x
]0, 1[
x
A-»
C
]0, aj+i[
x
]0, 1[
x
An-\n)
Composing the
embedding
affine
guaranteed by Proposition 4.6
obtain the
xi
°
aj+i
°
À7+i
4.6
i>,x,y)
=
Pj(u,v,x,y)
=
J
°
with the
symplectic
Tn(lJ+l)
diffeomorphism
->
Tj+i
symplectic embedding
Using Proposition
P;(m,
oXj+i:
°^T1i *f
Tn(l2+1)
^
and with the affine
Xj oaj+i
'=
Un~l(\)
diffeomorphism £7+i °X~\
<Pal+l:Saj+l
Pj
x
(^)
»-'
x
Sa]+X
—
we
(n4^j
=
(i)
°
we
<Pa,+i
°
Éj+i
°
x7+i
°
î+i
°
XJX
:
FJ+l
^
TJ+l-
find
(u,v,x,y)
if
u
<
2js +-rL^-n
(5.75)
'7+1
and
(u + s-kj+i-âj+i,v,jj±=jx,(.lj+i-l)y)
if
m
>
2js
+
T^(flj+i
-
w).
(5-76)
138
5. Proof of Theorem 3
In view of the
(pj (F;)
Pj
:
{ (m,
D
ft
°
identity (5.75)
y) |
v, x,
fj)
( LI $
extend the restriction of p; to the subset
identity to the symplectic embedding
the
2js ] by
>
m
we can
id
R>2n
P;
,
=
0;
on
Pj
.
finally
define the
Hj S,
t
.1=1
We
^
o
•
^7
0n
i
=
,
l
^7 (Sj + l)
°
symplectic embedding yjtj+i by
7+1
i/>;+i
:=
o^jofj: JJ
pj
R2".
S,
j=i
hypothesis (5.58)
In view of the induction
Pj(FJ+i)
Tj+i
C
we
7+1
7+1
JJS,
fJ+i:
from vJ+i
(5.57) and
=
Vj +
(5.62)
üJ+i
i=i
i=i
Moreover,
true.
7
]JF,(5)U]J/V,(5),
^
(=1
i.e., (5.60) holds
and Lemma 5.24 and the inclusion
have
(5.74)
deduce from formulae
we
and from the definitions of sJ+i,
tJ+i
and
(5.76),
and eJ+i given in
that
f]+i(u,v,x,y)
(u-vJ+i+Sj+i,v, J—^x,y/hJ+inyJ
=
if
i.e., formula (5.61) holds
true.
This
completes
M
>
Vj + i
-
Sj + l,
the inductive construction of the
symplectic embedding Vo+iComposing the linear symplectomorphism
ß
x
id2„_2
:
X
]J S,
i=i
given by
formula
-*
JJ S,
i=i
(5.56) with the inductively constructed symplectic embedding
k
k
fk:
\\S,
1=1
^
k-1
JjF,(5)UJJ^,(5)
1=1
1=1
5.2. Proof of lim
pf(M,
co)
=
1
139
a—>-oo
we
finally
obtain the
symplectic embedding
k
*a
:=
k
fko(ßxid2n-2):X\\Si
i=i
The
proof of Proposition
Step
6. End of the
We let
as
in
e
>
0 be
in Theorem
as
s
Using
we can
j=i
5.11,
D
5.11
e'
set
e/5,
—
choose k and ao
be the number associated with k and e' after the
=
ao(e)
proof
of
a
X(a) as in (5.51).
ao and define X
5.19, the identities (5.49) and the definitions (5.51) and (5.45)
Lemma 5.12. We fix
Lemma
j=i
complete.
proof of Theorem
3 and let
Step
5.20 is
k-1
WTi(8)v\\Ni(S).
^
>
=
estimate
\XTn(a,n)\
\X\\Si\-e'
>
=
k\XSi\-e'
=
kq | Fi |
e'
k±s2n-2e'.
=
Composing
—
(5.77)
the inclusion
k
La:XTn(a,n)
x\jst
^
i=l
symplectic embeddings tya and p guaranteed by Proposition
Proposition 5.16 we obtain the symplectic embedding
with the
Oa
In view of the estimates
:=
potyaoLa:
(5.42) and (5.77)
ß(M\®a(XTn(a,n)))
=
=
<
=
=
This is the
complete.
required
X
Tn(a, n)
we
^
5.20 and
M.
find
($a (X Tn(a, n)))
ß(M)-\XTn(a,n)\
ß(M)
-
ß
(^,2" 3,')-(^,2"-2e')
+
5e'
e.
estimate in Theorem 5.11
an so
the
proof
of Theorem 5.11 is
140
5. Proof of Theorem 3
5.3
Asymptotic embedding
Consider
again
volume Vol
connected 2n-dimensional
a
lim
(M, co)
one
symplectic
can
try
and
{
recapture
we
2«-dimensional
information
some
the convergence
there exists
s
such that 1
sup
In this section
pff(M,co)
lim
(5.78)
1
=
a-+oo
speeds
in
on
the geometry of
(5.78).
We define the
pff(M,
such that 1
a
notice that aE(M, co)
pff(M,
>
manifolds and
symplectic
<
oo
co)
<
Ca~s for a\\a
<
oo
<
Ca~s for all
constant C
—
£
1
C
a constant
-
there exists
ap(M, co)
co)
>
J
n
'
1
a
j
> n
'
>
£ for large classes of
thereby improve
Theorem 3 for many
or
ap(M, co)
manifolds.
Consider
and the
1
=
aE(M, co) and ap(M, co) in [0, oo] by
sup
=
study
to
invariants
oe(M, co)
symplectic
p!;(M,co)
In order to
uninteresting.
symplectic manifold (M, co) of finite
asymptotic symplectic invariants
In view of Theorem 3 the
(M, co).
a->oo
are
invariants
a
domain U in
boundary
R2".
The distance
d(p,dU)
between
a
point
p G U
dU is
d
(p, 3t/)
=
inf
{d(p, q) \
We say that U is very connected if there exists
e
dU].
q
G
>
0 such that the set
U\{peU \d(p,dU) <e)
is connected.
Theorem 5.25 Assume that U is
piecewise smooth boundary and
sional
(ï)e
(ii)E
(i)p
(ii)p
a
very connected bounded domain in
that
(M, co) is
symplectic manifold.
aE(U)
>
-
ifn <3orifU
aE(M,co)>l;
<rp(U)
>
I.
aP(M,co)>±.
ifn
<
3.
is
a
ball.
a
R2n
with
compact connected 2n-dimen-
5.3.
Rough outline of the proof of Theorem
from
6.6
Corollary
assertions
can
be
2
a
cube
proved by
Proposition
n
a
the
Assertion
symplectic folding
2n-cube. If
(i)^ for
a
2n-ball follows
n
=
methods
The other
previously
described.
2, assertion (i)^ follows from Propo¬
proved by using that in dimension
simplices. Assertion (i)p for a cube follows from
3, assertion (i)^
=
be filled with small
can
5.25:
(i) below which is proved by Lagrangian folding.
Assume first that U is
sition 3.16, and if
141
invariants
Asymptotic embedding
can
be
4.1.
very connected bounded domain in
R2"
with
pieceboundary. Then assertions (i)^ and (i)p can be proved by first ex¬
of connected unions of
hausting U with an increasing sequence Ui C U2 C
equal cubes and then extending the embedding techniques used to fill a cube to
Assume next that U is
a
wise smooth
...
the sets Ui.
Assume
finally
tic manifold.
finitely
that
(M, co) is
Then assertions
a
compact connected 2n-dimensional symplec¬
(ii)# and (ii)/>
many Darboux charts cpi
:
Ui
-*
V,-
can
be
proved by
C M such that the
Ui
first
are
disjoint
|J V,used
to prove
technique
and thiner neighbourhoods of the
are
the
=
very
con¬
piecewise smooth boundary and such that the Vi
connecting the Vi by lines, and finally applying
<Pi(U) and using thiner
(i)# and (i)p to the sets V,-
nected bounded domains with
and
choosing
M, then
=
lines to pass from
one
Vi
to another.
142
5. Proof of Theorem 3
6
Symplectic
6.1
versus
Lagrangian folding
Lagrangian folding
Lagrangian version of folding developed by Traynor in [34]. Here,
the whole ellipsoid or the whole polydisc is viewed as a Lagrangian product of a
cube and a simplex or of a cube and a cuboid, and folding is then simply achieved
by wrapping the cube around the base of the cotangent bundle of the torus via a
linear map. This version of folding has therefore a more algebraic flavour than
symplectic folding.
For the sake of brevity we shall only study Lagrangian folding embeddings of
skinny ellipsoids into balls and of skinny polydiscs into cubes.
There is
a
Theorem 6.1 Assume that
>
a
0 and that
ki
<
•
•
•
<
£„_i
are
relatively prime
numbers.
(i) The ellipsoid
E2n(n,
,2„/
Bm
for
any
e
>
I
...
a) symplectically embeds
,n,
.-
max
n,
P2n(n,
...
,n,
e
a) symplectically embeds into the cube
C2n ( max | kn-in, (n
VI
Theorem 6.1
(i) for
n
=
—
\)n
-\
h
2 is Theorem 6.4 in
=
{0
A(bi,...,bn)
=
|0<yi,...,y„
<xi
<
a,-,
=
U(a, ...,a) and An(b)
x
R"(y)
C
££<>
i
Un(a)
<n]
1 <i
D(ai,...,a„)
kn-i
[34]. We shall closely follow
[34]. We consider again the Lagrangian splitting R"(x)
and abbreviate
+
———
0.
(ii) The polydisc
Proof.
a
..
(*B_i + 1)
into the ball
=
=
F"=R"(x)/7rZ"
of
R2w,
set
R"(x),
C
R"(y),
l
A(b,..., b).
We also set
144
6.
Symplectic
versus
Lagrangian folding
and abbreviate
IC
Kl
=
•
•
•
iCn—l
and
Choose
folding
e
>
are
AE
=
max{(£„_i
Ap
=
max
We set e'
0.
E(n,...,n,a)
^>
{kn-in,
(n
f},
\)n
—
^}.
+
The
embeddings provided by Lagrangian
symplectic embeddings
e/Ag.
=
of
compositions
l)n,
+
"(!)
(1
x
+
e')A(n,...,
n,
a)
^'^'â+Â^l,...,^-!,^)
jn
^
x
(Ae±£\
An
B2n(AE
e)
+
respectively
—^
P(n,...,n,a)
"(!)
0(tv,
x
a)
n,
+
D(&'---'è^7r)xD(^'---'^-i'^)
%
Tn
4
C2n(AP).
x
(^f\
Un
1. The maps aE and ap. Recall from the
a
...,
proof of Lemma 4.4 (i)
that there exists
symplectomorphism
ai
x
x
:
an
P(n,
which embeds the subset
E(n,
We denote the reflection
(x, y)
mutation
(x, y)
h-*
(y, x) by
ap
symplectomorphically
restriction to
E(n,
maps
...,
n,
E(n, ...,n,a) into D"(l)
...
...
i-»
,n,a)
,n,a) into
(x,
a.
The
:=
a o
F(7r,
((1
of
...
,n,a)
x
ün(l)
e')A(n,
n, a)) x D"(l).
x
Rn(x)
R"(y) by p and the per¬
+
...,
composition
(ai
...
a), which
x
y)
—
0(n,
—>•
x
,n,a)
we
an)
x
to
o
D"(l)
denote
by
p
x
aE,
(1 + e')A(n, ...,n,a).
D(jr,... ,n,a), and its
symplectically
embeds
6.1.
145
Lagrangian folding
2. The map
given by
the
The map
ß.
is the linear
ß
of
symplectomorphism
R"(x)
x
R"(y)
matrix
diagonal
n
diag
ki
n
-
KJl,
-
.
,
.
.
,
ki
kn-i
,
.
.
.
,
,
n
3. The map y. In order to describe the
1
kn—i
,
icn
map y
we
wrapping
n
need
an
elementary
lemma.
Lemma 6.2 Let M
:
R"(x)
—»
R"(x) be
the linear map
l\
given by
the matrix
^
-k
0
i
kn-\
1
and let p:
Rn(x)
poM: Rn(x)
Proof.
Let
x
-+
Rn(x)/nlll
—>
Tn embeds
and x' be
points
Tn be the
=
projection.
The
composition
d(^,..., -£-, Kn\ into Tn.
( jf-,...,
in
-p—,
icn
)
for which
(p
o
M)(x)
=
(poM)(x').Then
I <-,
x,
X,'
|
î
I
<
i
,
=
1,
...
,n
—
1,
(6.1)
and
and there exist
integers h,..., Z„_i
xi
and
an
integer /„
k,
the
icn,
(6.2)
k" +li7t
(6.3)
such that
Xi
—
such that
xn
Inserting
<c
Xfi
xn
identity (6.4)
—
(6.4)
xn + inn.
into the estimate
\ln\
<
(6.2)
K
we
obtain
(6.5)
146
6.
and
inserting (6.4)
into the identities
x'i-Xi
The identities
I,
,n
...
ln
that
=
—
(ft li)jt,
/„ ^ 0.
_i_
/.
<
i
F,
Then the estimates
I. Since the numbers ki,...,
mki
kn-i
in contradiction to
li
=
Lagrangian folding
obtain
i
+
=
for
some
=
=
(6.6)
l,...,n-\.
The restriction of p
o
=
0, and
M to D (
so x
=
=
(6.7)
\,...,n-\.
(6.7) imply that /„
kn-i
are
0. In
ki ---kn-i
>
(6.5). Therefore /„
/„_i
=
integer m ^
\ln\
also
we
versus
(6.6) and the estimates (6.1) yield
k
Assume that
(6.3)
Symplectic
=
+
fc,7,-
relatively prime,
particular,
0 for i
=
we
=
conclude
K
0. The estimates
(6.7)
imply
now
that
x' in view of the identities (6.4) and (6.3).
|-,..., ^—,
ten) is therefore injective,
as
claimed.
D
We denote
by
M* the transpose of the inverse of M. Then the map
M
is
a
x
M*: Rn(x)
symplectomorphism.
linear
x
Rn(y)
-
R"(x)
by
y
=
(p
o
M)
x
R"(y)
In view of Lemma 6.2 the map
>/:D(ê'---'ér'/C7r)xR"(>;)
defined
x
M* is
a
->
TnxW(y)
symplectic embedding.
Lemma 6.3 We have
y(D(&'---'è'^)x(1 >(^---'*»-»'^))c7',,xA,,(ii?£)
+
and
^
(D(ïT'-'3^7'^) xD(^--*»-^ ^)) CF"XD«(^).
6.1.
Lagrangian folding
Proof.
147
In view of the definition of y and since
(1
e')As
-I-
Ae +e it is enough
=
to show that
M*(A(*i,...,^_1,^))cA»0f)
and that
uä)cü"(^).
^(q(*i
We
compute that M* is given by the matrix
\
/l
0
0
Ui
k2
kn-\
(ki,..., kn-i, ^) and set y'
definitions of A (ki,..., kn-i, -^) and Ae
then find
We
assume
first that y
A
g
=
M*y. Using
the
we
/
yi
i
+
,
---
+
i
/z
yn
=
1\
i
yi
(ki + i)-
i
n
i
+
---
i
yn-1
1\
kn-i
ki
<
max|A:n_i
I
+
a
.
(kn-i + i)-— +
+
—
\cn
yn
—
—
1,—}
Kn
J
n
Therefore
We
y'
G
assume
definitions of
A"
(^f- j
as
claimed.
(ki,..., kn-i, -^)
next that y G
(ki,
...,
kn-\,
f^)
and
Ap
we
y'GD(*1,...,fc„_1,„-l
as
+
and set
y'
=
the
^)cD"(^)
claimed.
4. The maps
M*y. Using
then find
D
8e and Sp. We define symplectic embeddings
8E
:
Un(7t)
x
An
(^)
<-»
B2n (AE
+
e)
148
6.
Symplectic
Lagrangian folding
versus
and
8P
:
nn(n)
Dn
x
(àf\
*-+
C2n (AP)
by
(xi,
...,yn)
...,xn,yi,
(y/yïcos2xi,...,^/yïcos2xn,
-^/yl sin 2xi,
-V^sin2x„)
h+
•
The
embedding 8E
extends to
a
extends to
embedding Sp
8p: Tn
,
ßj^-)
a
^
B2n(AE
+
e)
symplectic embedding
xDn(àA
^
C2n(AP).
completes the construction of the symplectic embeddings
grangian folding, and Theorem 6.1 is proved.
This
Assume that
As in the
a
>
0 and that
proof of Theorem
:
ki
we
<
•
=
max
Ap(a;ki,...,kn-i)
=
max
]2n, oo[
-»
we are
R defined
<
kn-i
are
)(fc„-i 1)
|fcn-i7T, (n
+
relatively prime
n,
-
ky.akn_x
kv.akn_^ \
i|"5
]2n, oo[
inf
{Aß (a; ki,..., kn-i) \ki
<
l2PC(a)
:=
inf
{Ap (a;ki,..., kn-i) \ki
<
n
=
the minimum. We shall next
2 and
n
=
3.
->
R and
•••
<
kn-i
are
relatively prime},
<
kn-i
are
relatively prime}.
Notice that the infimum in the definitions of the functions
for
:
by
:=
lpnc
numbers.
j,
\)n +
interested in the functions
l2Ep(a)
replaced by
involved in La¬
abbreviate
AE(a;ki,...,kn-i)
In view of Theorem 6.1
l2pnc
6.1
.
symplectic embedding
8E:Tnx An
and the
•
l\fB
explicitly compute the
and
l2pnc
functions
can
l2EnB
be
and
6.1.
Lagrangian folding
149
6.4 Assume that
Corollary
>
a
2n.
(i) The ellipsoid E(n, a) symplectically embeds into the ball B4
for
any
e
(k
hß(a)
if (k
\)n
+
f
\)(k
-
if k(k
1)
+
+
1)
<
<
f
<
f
<
k(k
k(k
+
(ii) The polydisc P(n, a) symplectically embeds into the cube C
\
kn
\
\+n
A
lPC(a)
PCK
=
(k-\)2
<
if (k-l)k
<
if
Proof, (i) follows from Theorem 6.1 (i), from
lEß(a)
and from
=
the
f
<
*
~
%
<
1),
+
2).
(lpc(a))
where
(k- \)k,
\
k2.
identity
=
min
the
max
identity
{kn, f+ n)
straightforward computation.
a
Corollary
The
e)
a
l4pr(a)
(i)
+
minmax{(Ä:+l)7r, |}
straightforward computation.
(ii) follows from Theorem 6.1 (ii), from
and from
(/^5(a)
0 where
>
6.5 Assume that
ellipsoid E(n,
for any
IebW
e
>
n,
a
>
2n.
a) symplectically embeds
into the ball
B6
(l%B(a) + e)
0 where
(k
+
\)n
if (k
-
2)(k
-
l)(k
+
1)
<
£
<
(k
-
\)k(k
+
1),
=
jj^
(ii) The polydisc P(n,
n,
if (k-\)k(k
+
\)<^<(k-\)k(k
2).
+
a) symplectically embeds into the cube C6
(lPC(a))
where
6
lpc{a)
_J
(k+l)n
if
(k-\)2k<%
<(k-\)k(k+\),
~
j
W+T)
+ 2yr
V (k
~
Vk(*k
+
!)
<
f
^
kl(~k
+
V-
150
6.
Proof,
Fix
(i)
Then &2
>
a
1 and
—
&2
2n and
are
also
assume
that
Symplectic
ki
relatively prime
are
numbers.
and
relatively prime,
AE(a;k2- l,k2)
&2
<
Lagrangian folding
versus
AE(a;ki,k2).
<
It follows that
4ß(a)
min
=
max
I (k +
\)n,
j^fA
Assertion (i) follows from Theorem 6.1 (i), from the
(6.8)
.
identity (6.8)
and from
a
straightforward computation.
(ii)
The
same
argument
l%r(a)
Assertion
as
above
implies
that for each
2n,
>
a
=
min
max
\ kn,
=
min
max
| (k + l)n, k{£+l) +2n\.
2n +
(ii) follows from Theorem 6.1 (ii),
,.
\
a,s,
from the
(6.9)
identity (6.9)
and from
a
straightforward computation.
For
n
>
do not know
describes the
Corollary
4 the
computation
6.6
(i)
There exists
(ii) There exists
a
a
constant
We choose
be the least
n
common
constant
is
far all
>
1
-
"
Cp a-l'n
1
1

corollary
2.
on n
>
we
such that
2n.
such that
for all
prime numbers pi <
multiple of the differences
—
on n
involved, and
more
Ce depending only
Cp depending only
Pj -Pi,
>
lpnc(a)
"
|C2"(/2"c(a))|
Fixa
and
>\-CE a~1/n
'ffr';-:'*'??1
to
l2EnB(a)
explicit formulae for the functions l2EnB and l2pnc.
asymptotic behaviour of these functions for all n
'ffr'"J'*'??1
I a2" (/&(«))!
Proof, (i)
of
•
a
•
•
>
<
2tt.
Pn-i and define /
1.
2n. We set
m
:=
i(ë+'"'-'
(6.10)
6.1.
Lagrangian folding
151
where |>1 denotes the minimal
integer
which is greater
ki:=ml-pn-i,
We claim that the numbers
assume
some
i
I ml
—
so
<
•
•
•
to r, and
(6.11)
\,...,n-\.
=
kn-i
<
-
d divides the least
and
pi
/ j. Then d divides
(ml
and
^2
<
equal
are
relatively prime. Indeed,
that
d
for
ki
i
or
d
\ ml
(6.12)
pj
—
the difference
Pi)- (ml
common
pj)
-
pj
=
/.
multiple
-
pi
But then
(6.12) implies
that d
1 as claimed.
divides both p,- and pj, and so d
Using the definitions (6.11) and (6.10) we find
=
fa~
ml-pn-i
>
J-.
(*„_! +l)fcn_i---*i
>
tf
ki
=
Therefore,
>
-.
n
We conclude that
(kn-i
and
l)n
>
ki ••k„-i
so
(kn-i
By
+
definition
+
(6.10)
\)n
we
=
AE(a;ki,...,kn-i)
>
/|"ß(a).
have
m,il(i(ß+"^) 1) ß+p"-^L
+
Using
definition
=
ml-pi
(6.13) and (6.15)
\B2n (l2EnB(a))\
\E(n,
...,
=
(6.11) and the estimate (6.14)
kn-i + l
The estimates
n,
a)\
(6.13)
^
(kn-i
now
+
a
+ l
<
we
find
y^
+
pn-i+l.
yield
l)nn
^
(6-14)
(yg-+(pn-i+l)l/W)n
a
(6.15)
152
6.
Recall that the number
(6.16)
we
(pn-i
therefore find
a
+
and
n,
...,
versus
Lagrangian folding
I) tfn only depends on n. In view of the
Ce depending only on n such that
estimate
constant
I a2" (#*(*))!
\E(n,
Symplectic
CE a~1/n
for all
a
>
2n
>1-CE a-""
for all
a
>
2n.
1 +
<
a)\
so
'f?';;'*'??1
Assertion
(ii)
(i) is proved.
can
6.2
be
proved
in the
Comparison
plectic
Recall from
embedding
and
same
of the
way
as
(i).
embedding
results obtained
by
sym¬
Lagrangian folding
4 and 5 that
symplectic folding yields good symplectic
ellipsoids and polydiscs into any connected
symplectic manifold of finite volume. The reason is that symplec¬
Chapters 3,
results of 2n-dimensional
2n -dimensional
folding is local in nature in the sense that each fold is achieved on a set of small
volume. Lagrangian folding described in Section 6.1, however, is global in nature
and thus yields good symplectic embedding results of ellipsoids and polydiscs
into special symplectic manifolds only. E.g., the best 2«-dimensional Lagrangian
tic
folding embedding of an ellipsoid into a cube respectively of a polydisc into a ball
fill less than ^ respectively ^ of the volume.
In this section we shall compare the embedding results for embeddings of
skinny ellipsoids into balls and skinny polydiscs into cubes yielded by the two
methods.
6.2.1
Embeddings £2n(7r,...,
1. The
case n
=
plectically
embed into the ball
tions seb
]2n, oo[
and defined in
F(7r, a)
folding.
^->
a)
^
B2n(A).
2.
Recall from Theorem 1 that for
'
n,
—»
R and
Corollary
B4(A)
6.4
obtained
(i)
a
G
[n, 2n] the ellipsoid F(7r, a) does
B4(A)
Ieb
=
if A
1%b
describe
'•
our
<
2n.
^n' °°[
not sym¬
Also recall that the func¬
~~*
^ constructed in 3.3.1
results for the
embedding problem
and by Lagrangian
by multiple symplectic folding
6.2.
of
Comparison
According
Proposition
to
and
symplectic
Lagrangian folding
3.8 the difference
the volume condition is bounded
by
difference is monotone
and
Ieb(o)
k(k
+
—
\pna
increasing
\)n, where it is equal
The sequence
k(k
2)n,
+
For
a
S£ß(47T)
=
we
2.6916
to mk
equal
|-,
to
to
and
+Jk(k
Mk
n, and
+
<
2)n
on
an
k(k
+
seb(o) happens first
2)n;
moreover,
lim
k—^-oo
\)n
and
we seem to
have
k(k
+
(sEB(k(k
+
g
sEß(a)
—
a
and
Ieb(o)
=
embedding problem E(n, a)
/£ß(a)
The
=
inf
I
A
following proposition
N
at
a
into
for
°->-
<
2)n)-lEB(k(k
<
a
B4(A)
Ieb(o)
2)n))
+
is the function
summarizes what
Figure
53. In
<
n
=
fsB(a)
general,
yield
and seb
Ieb(o)
an
and
<
seb(o)
interval around
0.
[n, oo[ by setting
optimal function for the
fßß on [n, oo[ defined by
The
we
<
B4(A) \.
know about this function.
Proposition 6.8 For a G [n, 2n] we have fEß(a)
function fEB is bounded from below and above by
see
on
In
.n.
Ieb
to functions on
[n,2n].
G
to have
| E(n, a) symplectically embeds into
max(2n,*/na)
2.3801...
5 000.
seb(o) and Ieb(o)
a
=
54(2.381:r),
5.1622..
=
we seem
seb(o)
We checked the above statements for k
We extend both functions
2)n.
to n. We
so
better estimates: For all k
interval around
+
—
the computer calculations for seb suggest that the functions
alternately
.s/k(k
—
(Mk) strictly decreases
ellipsoid E(n, 3n) symplectically embeds
E(n, 4n) symplectically embeds into B4(2.692n).
inequality Ieb(o)
at
\)n, and it attains its
+
(k
—
Ieb(o)\ is bounded by 2n.
sEß(a) < lEß(a)- E.g., sEß(3n)
\seb(g)
have
...
(k + \)n
—
Fact 6.7 The
The
between seb and
Computer calculations suggest that this
converging to n as a —> oo. The difference
=
(mk) strictly decreases
2n small
>
-Jna
—
2n.
where it is
conclude that the difference
and
seb(®)
Ieb and the volume condition attains its local minima
between
local maxima at
153
=
a,
and
on
]2n, oo[ the
min(/£ß(a), sEß(a)),
particular,
v
fEB(2n
+
hmsup
e^0+
e
e)-2n
3
<
-.
7
(6.17)
154
6.
The function
is
fEB
increasing and hence almost everywhere differen¬
Lipschitz continuous with Lipschitz constant at most 1;
min
-
fEBifl)
<
(Ieb(o), seb(ci))
Proof. The estimates of fEB (a) from below
Hofer
>
Lagrangian folding
precisely,
fEß(a')
a
versus
monotone
tiable. Moreover, fsB is
more
Symplectic
which
capacity,
2n, and by
lates to
fEB (a)
Proposition
yields fEß(a)
>
the volume condition
>
a
(a'
are
for
a
a)
—
for all
provided by
G
\E(n, a)\
[n, 2n]
> n.
the second Ekeland-
and
/gß(a)
\B4 (/eß(a))|,
<
> a
a
>
2n for
which trans¬
+Jna~. The estimate (6.17) follows from fEB
seb and
claims
follow
in
the
of
as
remaining
proof
Proposition
<
3.6. The
from
3.12.
D
2n
An
6n
8n
Figure
2. The
case n >
\2n
I5n
20n
53: What is known about
24n
/£ß(a).
3.
Recall from Theorem 1 that for
a
G
[n, 2n] the ellipsoid
E2n(n,
...
,n,
a) does
symplectically embed into the ball B2n(A) if A < a. For a > 2n, the ra'th
capacity still implies that E2n(n,... ,n,a) does not symplecti¬
cally embed into B2n(A) if A < 27r. This information is vacuous if a > 2n7T in
view of the volume condition A > \/nn~xa.
not
Ekeland-Hofer
6.2.
of
Comparison
If
>
a
E2n(n,
For
a
for
n
we
the results obtained
R defined before
Corollary
tween the function
bounded
—
(|
by symplectic folding,
3 and the first statement in Theorem
by Lagrangian folding,
which
6.4.
are
For
3,
described
n
are
n
are
for every
0.
>
e
Theorem 5.25
(i)^
weaker than the results ob¬
3 the difference
=
e)
+
which
by the function l2EnB
lEB(a)
:
—
]2n, oo[
\ln2a
->
be¬
in
and the volume condition is bounded.
6.2.2
Embeddings P2n(n, ...,n,a)^ C2n(A).
1. The
case n
2.
Recall that the functions spc
defined in
+
shows that
Corollary 6.5 (i) and the volume condition is
follows from Corollary 6.6 (i) that the difference
lEB computed
by 27r. For n > 4 it
y/nn~1a between l2EB
=
155
proved by symplectic folding,
,n,a) symplectically embeds into B2n
...
=
l2EB(a)
Lagrangian folding
2n, Theorem 2.1, which
large,
tained
and
symplectic
]2n, oo[
:
Corollary
3.18 and
Proposition
—>
R and
6.4
Ipc
lPC : ]2n,
=
(ii) describe
our
oo[
R
->
results for the
em¬
bedding problem P(n, a) °-» C4(A) obtained by multiple symplectic folding and
by Lagrangian folding.
Comparing Proposition 3.18 with Corollary 6.4 (ii) we see that lpc(a) <
spc (a) for all a > 2n, cf. Figure 54. Equality only holds for a g ]2n,4n[
and
a
e
[k2n, (k2
+
l)n], k
>
2.
The difference
and the volume condition attains its local minima at
equal
is equal
to mk
to
=
n.
kn
+/k(k
—
The sequence
Extend the function
a G
fpc(a)
—
y/na
\)n, k
>
\)n, and it attains its local maxima at
(mk) strictly decreases
to
between
fpc
=
on
inf
6.9 The
54. It is
tiable. Moreover,
|A
|-.
| P(n, a) symplectically embeds into
summarizes what
function fpc'- [n,oo[
monotone
fpc
where it
c-^
=
a
for
C4(A)
[n, oo[ defined by
4
Figure
Ipc
3, where it is
k2n,
[n, oo[ by setting lpc(a)
to a function on
lpc(o)
following proposition
Proposition
above by
see
k(k
—
[n, 2n]. The optimal function for the embedding problem P(n, a)
is the function
The
—
lpc(a)
is
<
fPC(a)
increasing
Lipschitz
we
—»
<
C4(A) \.
know about this function.
R is bounded from below and
I pc (a),
and hence almost
continuous with
everywhere differen¬
Lipschitz
constant at most
1;
156
6.
more
Symplectic
versus
Lagrangian folding
precisely,
fpc(a')
fpc(a)
—
(a'
<
—
for ail a'
a)
> a
> n.
a
The estimate
Proof
dition
| P (n, a) |
Proposition
«Jna~
fpc(a)
<
| C4 (fpc(a))
<
\
from below is
The
•
provided by
claims follow
remaining
the volume
as
in the
con¬
proof of
3.12.
/
/A
5n
^pr(a)^^^^
3n
/^îp^^
-J^Â-^
An
2n
>
6n
9n
54: What is known about fpc(a)-
Figure
case n
a
An
2n
2. The
=
3.
Recall that the functions
spnc :
]2n, oo[
->•
R and
l2pc :
]2n, oo[
->•
R defined in
Proposition 4.1 and before Corollary 6.4 describe our results for the embedding
problem P2n(n,... ,n,a) ^ C2n(A) obtained by multiple symplectic folding
and by Lagrangian folding.
3 of Proposition 4.1 with Corollary 6.5 (ii) we see
Comparing the case n
=
that in contrast to the
Equality only
difference
d3
=
holds for
sPC(a)
—
a
e
</n2a
=
[((k
2
-
we
have
\)k2
+
sPC(a)
2)
n,
(k
<
-
lPC(a)
l)k(k
+
for all
\)n],
k
2n.
a
>
>
3. The
between sPC and the volume condition is bounded
by
(i3--#1586)tt.
For
we
case n
n
>
A the
do not know
comparison
an
explicit
of the functions
formula for
l2pc.
s2pc
and
lpnc
is
The difference
more
involved since
s2pc(a)
—
4nn~xa
6.2.
Comparison
between
d4
=
s2pc
symplectic
(5-^194)^,
(ii)
(dn),
d5
n
>
that the difference
difference
\s2PC
—
l2PC\
symplectically
dn
3, strictly decreases
l2PC(a)
—
\lnn~xa
by d„
=
to
157
h
n.
where
-
^2n~l+2\n,
It follows from
is also bounded.
n
>
6.
Corollary
Therefore, the
is bounded.
We conclude this section
Theorem 1.3
Lagrangian folding
(4-^164)n,
=
of Section 1.3. We say that
it
and
and the volume condition is bounded
The sequence
6.6
of
by motivating the conjecture alluded to at the end
,n,a), a > n, is reducible if
polydisc P2n(n,
a
...
embeds into
(ii) in [23] that
a
cube
C2n(A)
for
some
A
<
a.
It follows from
polydisc cannot be reduced by a local squeezing
symplectic embedding which reduces P2n(n,... ,n,a) must therefore
be of global nature. In view of Proposition 4.1 and Corollary 6.4 (ii) symplectic
and Lagrangian folding both show that P2n(n,
n, a) is reducible if a > 2:zr.
However, none of the two folding methods can reduce P2n(n,..., n, a) if a <
2n. Since we believe that only some kind of folding can reduce a polydisc, we
conjecture
a
method. A
...,
Conjecture 6.10 The polydisc-analogue of Theorem 1
the polydiscs P2n(n,
,n, a) symplectically embeds
some A < a if and only if a > 2n.
...
holds
true.
In
into the cube
particular,
C2n(A) for
158
6.
Symplectic
versus
Lagrangian folding
7
Proof of Theorem 4
Throughout this section, c will denote a normalized extrinsic symplectic capac¬
ity on R2 as defined in the introduction. We refer to Appendix B.2 for results
about such capacities and only mention that any normalized intrinsic symplectic
capacity on R2 as defined in Definition B.l is a normalized extrinsic symplectic
capacity on R2.
symplectic embedding of an arbitrary subset S of R2" into another subset
S' of R2" is by definition a symplectic embedding of an open neighbourhood of S
into R2n which maps S into S'. As before we denote by Z2n(a) the open standard
symplectic cylinder D(a) x R2"-2, and we denote by Ez C R2n the affine plane
A
Ez
Given any subset F of
c
Assume
now
define the
(T
n
:=
Z2n(n)
Ez)
:=
c
R2
we
x
[z],
z g
R2"-2.
abbreviate
({(u, d)êR2|
(u, v,z)
R2w symplectically
[0, n] by
ijc(S)
that the subset S of
symplectic
invariant
ÇC(S)
:=
over
all
embeds into
Z2n(n).
infsuVc(cp(S)nEz)
V
where cp varies
Ez}\.
T n
e
We
(7.1)
z
symplectic embeddings
of S into
Z2n(n).
The main result
of this section is
Theorem 7.1 Assume that the subset S
for some
c on R2.
a
< n.
Assume
now
plectomorphism
[0, n] by
Then%c(S)
that S is
of
R2n.
a
=
0 for any normalized extrinsic
subset of
:=
inf sup
f
now
cp varies
R2"
over
all
Z2n(n).ThenÇc(S)<Çc(S).
c
(cp (S)
symplectic
n
into
Z2n(a)
symplectic capacity
which embeds into
We define the ambient
Çc (S)
where
ofR2n symplectically embeds
Z2n(n) by
invariant
a
sym¬
ÇC(S)
e
Ez)
z
symplectomorphisms
of
R2"
which embed S into
160
7. Proof of Theorem 4
relatively compact subset ofR whose clo¬
Then £c (S)
0 for
sure embeds into Z2n(n) by a symplectomorphism o/R2".
R2.
extrinsic
c
on
normalized
symplectic capacity
any
Corollary
7.2
(i) Assume that S is
a
=
(ii)
S1
For the unit circle
R2
C
have £c
we
normalized extrinsic symplectic capacity
c on
(B2n(n))
G
[c(Sl), n]for
any
R2.
Remarks 7.3
1. Theorem 4 of the introduction is
explain
in Remark 7.8.2 below how
unbounded subsets of
special case of Corollary 7.2 (i). We shall
Corollary 7.2 (i) can be extended to certain
a
R2".
0 for any intrinsic symplec¬
According to Corollary B.9 (i) we have c(Sl)
tic capacity c on R2 and so the statement in Corollary 7.2 (ii) is empty for these
capacities. On the other hand, Proposition B.l 1 says that for any a e [0, n] there
exists a normalized extrinsic symplectic capacity c on R2 such that c(S1)
a.
R2
with
of
normalized
extrinsic
on
n
c(S1)
symplectic capacities
Examples
are the first Ekeland-Hofer capacity [6, Theorem 1], the displacement energy [17,
O
Theorem 1.9] and the outer cylindrical capacity z, see Theorem B.7 (iii).
2.
=
=
=
assumption there
Since S is
A
>
(i)
Fix
We denote the closure of S
S.
By
symplectomorphism cp of R2" such that cp (S) C Z2n(n).
compact, cp(S) is compact in R2". We therefore find a g ]0,n[ and
Proof of Corollary
7.2.
exists
0.
e
>
C
D(a)
by
a
0 such that
cp(S)
Choose a'
G
x
B2n-2(A).
(7.2)
]a,n[. Then
D(a)
x
B2n-2(A)
In view of Theorem 7.1 there exists
f:
a
C
Z2n(a').
(7.3)
symplectic embedding
Z2n(a')
^
Z2n(n)
(7.4)
such that
sup
c
if (z2n(a')\
Applying Proposition C.l to the bounded
find a symplectomorphism ^ of R2" such
*Id(û)xB2"-2(/1)
=
n
Ez\
<
(7.5)
e.
starlike domain
D(a)
x
B2n~2(A)
we
that
ir\D(a)*Bln-1{A)-
C7-6)
161
The inclusion
(7.2), the identity (7.6) and
(*o<p)(S)
In view of the inclusions
embeds S into
Z2n(n).
symplectic capacity
sup
c
((*
We conclude that
c
o
the inclusion
(7.3) show that
f(z2n(a')y
C
(7.7)
(7.7) and (7.4) the symplectomorphism
*P
o
cp of
R2"
Moreover, the inclusion (7.7), the normalization of the
and the estimate
cp) (S)
ÇC(S)
Ez)
n
<
Çc(S)
<
(7.5) yield
sup
(f (z2n(a')\ fl£z)
Since
e.
<
c
e
0
>
<
arbitrary,
was
e.
assertion
(i)
follows.
(ii)
We abbreviate
Slz
Let cp be
cording
a
{(m,
:=
z)
v,
symplectomorphism
1.2 in
to Lemma
a
R2n
of
+
=
l),
ze
R2"-2
zo £
B2n(n)
of
R2n
and cp
(B2n(n))
C
Z2n(n),
is tangent to the
boundary Sx
.
an e
Ae
is contained in cp
:=
{(u,
(B2n(n))
v,
n
rS^
In view of the
conformality
x
the
R2n~2
of
boundary
Z2n (n)
=
c(rSl)
G
1
EZ0 |
For each
A6
and the
=
c
at
and the compactness
C
cp
r
-
g
e
<
]1
—
(B2n(n))
monotonicity
\(u, v)\
e,
n
1[
<
we
l}
then have
Fz.
of the
symplectic capacity
obtain that
r2c(Sl)
Ac¬
>
zo)
EZo.
C
Z2n(n).
into
such that
(cp (B2n(n)))
(B2n(n))
each point of S\
This, the inclusion cp (B2n(n)) c Z2n(n)
of S\ imply that there exists
0 such that the annulus
of cp
9
R2n~2.
d(cp(B2n(n)))f\EZ0.
C
diffeomorphism
v2
which embeds
[23] there exists
Si
Since cp is
Ez \ u2
e
(rS^)
<
c(Ae)
<
c
(cp (ß2n(n)) Ez)
n
c we
162
7. Proof of Theorem 4
for each
r
g
]1
—
e,
1[, and
so
c(^(52"(7r))nFz)
Since this estimate holds for any
B2n(n)
into
Z2"(7r)
symplectomorphism
<
7.1. We fix
1. Reduction to
Proposition
a
the
so
Corollary
normalized extrinsic
a
c(D(n))
=
7.2 is
n.
=
complete.
symplectic capacity
c on
R2.
problem
7.4 Assume that
(z4(a)\
Then Theorem 7.1 holds
Let S be
a
for all
0
=
G
a
]0, n[.
true.
subset of
R2"
for which there exist
a
<
embedding cp: S °-^ Z2n(a). We fix e > 0. By assumption
embedding f : Z4(a) ^ Z4(n) such that
sup
zeR2
The
which embeds
c(Sl).
of
proof
4-dimensional
Çc
Proof.
>
supc(B2n(n)nEz\
(ii) thus follows, and
Proof of Theorem
Step
R2n
hand, the definition of Çc and the normalization of c yield
Çc(B2n(n)\
Assertion
cp of
conclude that
we
Çc(B2n(n))
On the other
c^1).
>
(V (z4(a))
c
V
V
n
Ez)
<
n
and
a
symplectic
we
find
a
symplectic
(7.8)
e.
/
/
composition
p: S
-^ Z2»
=
Z4(a)
symplectically embeds S
plectic capacity c and the
sup
zeR2"-2
c
(p(S)
fi
x
into
Z2n(7r).
estimate
Ez)
*î=U Z4(n)
R2"~4
=
Z2n(n)
(7.8) yield
sup
zeR2"-2
c

arbitrary,
was
conclude that
we
ÇC(S)
2. Reformulation of the 4-dimensional
Step
We shall
e
0
use
coordinates
0. We may
>
u, v, x, y on
assume a
the
0<u<n,0<v<\}
Figure
A
Z
:=
{(u,
v, x,
:=
Rx
R2.
Proposition
y) \
e
du Adv + dx
(n
<
u,
a
<
0
<
7.5 Assume there exists
a
a)/9.
—
hull of the
rectangle
v
I
<
D
problem
e
convex
58 below. The domain R is
claimed.
as
(R4,
and
n/2
>
0,
=
A
image
e, 0
< x
by
a
R c
< n
and
{ (m, v) \
of the map y drawn in
with smooth
—
Fix
dy).
Denote
corners.
<
1, 0
symplectic embedding
4>
We set
<
y
A
:
<
^
a},
Z such
that
sup
zeR2
Then there exists
(<5>(A)
n
symplectic embedding
a
sup
zeR2
Proof.
c
c
^
(* (z4(a))
V
V
Ez)
:
n
'
2e.
<
Z4(a)
Ez)
(7.9)
°->
<
Z4(7r)
such that
2e.
(7.10)
'
We start with
Lemma 7.6
(i) There
exists
a
a :
(ii)
There exists
a
a :
(iii)
There exists
a
symplectomorphism
R2
->•
{ (m, v)
g
R2 |
e
u,
0
< x
give
an
explicit
<
v
<
1
-
e
}.
symplectomorphism
D(a)
->
{(x, y)
e
symplectomorphism
R2 |
co
ofR2
Proof, (i) follows from the proof of Lemma 2.5
we
0
<
<
1, 0
such that
or
from
construction. Choose orientation
<
y
co(R)
< a
C
}.
D(n).
Proposition A.6 (ii). Here
preserving diffeomorphisms
164
g
7. Proof of Theorem 4
R
:
]0, oo[ and /z
—>•
:
R
—>
x
R,
]0, 1
—
e[, and denote by g' and h! their derivatives.
Then the maps
ai
:
R2
a2
:
]0, oo[
]0, oo[
-+
R
x
]e, oo[
-*
symplectomorphisms.
are
The map
(ii) follows from Lemma 2.5
(iii) Choose
phism
x of
G
r
D(2nr2)
R
]0, 1
x
or
i->
U(m),
(m, v)
h*
(-^ +
ai is therefore
a :=
«2
from
Proposition A.6 (i).
that R
large
so
which is
e[,
-
o
D(nr2),
c
translation
p^yj
(m, u)
as
e,
origin,
identity near
h(v)J
desired.
and choose
the
,
a
diffeomor¬
maps the
boundary
boundary of
D(2nr2). By Lemma 2.5 and its proof, we may assume that x is a symplectomor¬
phism. Extend x to the symplectomorphism of R2 which is the identity outside
D(2nr2), and let co be the inverse of this extension. Then co(R)
D(area(R)) C
of
to the
D(area(R))
a
boundary
near
of R, and is the
the
=
D(n),
desired.
as
Z be
symplectic embedding as assumed in the proposition,
let a, a and co be symplectomorphisms as guaranteed by Lemma 7.6, and denote
the linear symplectomorphism (u, v, x, y) h* (x, y, u, v) of R4 by r. Then the
composition
Let $
:
A
<->
,t,
*I>:
is
a
a
-7-4/
Z
(ax°OoT
x
>
(a)
symplectic embedding. Moreover,
imply
Z
coxid
the
4
>•
Z
x
a)
(71-)
(x (Z4(a)))
=
A, the
the assumed estimate
(7.9)
that
sup
zeR2
(v (z4(a))
c
V
V
n
'
Ez)
=
sup
zeR2
=
sup
zeR2
'
<
This
completes
Step
3. Construction of the
We
—>
identity (a
symplectic capacity c on R2 and
of the
monotonicity
<t>
A
are
going
estimate
the
on
of
Proposition
to construct a
(7.9) by
ternatingly
proof
the
a
and
((co
x
id) (<Ï>(A))
n
Ez)
c(<$>(A)Ç\Ez)
2e.
7.5.
embedding
$
symplectic embedding <ï> : A *-+ Z satisfying the
multiple symplectic folding. Instead of folding alon the left, we will always fold on the right. If we
variant of
right
c
165
neglect
all
quantities which
as in Figure 55.
will look
be chosen
can
arbitrarily small,
the
image <Ï>(A)
c Z
2a-
Figure
The
cp
and y
55: The
embedding
symplectic embedding
x
$
:
.>4>
<—>•
Z for
composition of the
preserving embedding
area
<
m, 0
<
is similar to the map constructed in
Step
1 of Section 3.2.
map
ß
black
{ (m, v) I
:
restricts to the
region
in
{2e
identity
< u
< a
ß(u, v)
for
m
G
The
]e + i2e,
"lifting"
Indeed, choose
e
+
(i
+
e
=
v
<
1
—
e
{e
ß(u
l)2e]
off function
Indeed,
<

[0, 1
—
e
—
such that
à
three maps
R2
}
see
Figure
map cp is similar to the map constructed in
a cut
^
2e), and it maps {2e <
9e} drawn in Figure 56. Moreover,
on
+
3tt
=
<î> will be the
id. Here, the smooth
ß
a
\ fo(s)ds
Jtr.
>
a.
Step
56.
3 of Section 3.2.
8] with support [3e,a + 8e]
166
7. Proof of Theorem 4
I
e2e3e
1
-e
e2e
a
f(s)
f0(s -i(a
=
Se)),
+
symplectomorphism cp : R4
cp(u,
v, x,
n
56: The left part of the
Figure
Set
+ 9e
y)
=
R4
-+
I
=
i
1, 2, 3,
is defined
and
...,
of
ß.
f(s)
=
£,.>0 f(s).
The
by
f(u)x,
m, i> +
image
\
x, y +
f(s )ds
(7.11)
JO
The
image
tion
(m,
equal
cp
v, x,
to the
((ß
y)
x
h->-
id)(A))
C
(m, v) by
R4
is illustrated in
Figure
57. Denote the
p. The left part of the set p
(cp ((ß
x
projec¬
id)(A))) is
upper domain in
Figure 58.
cp((ß x id)(A))
preserving
embedding y : p (cp ((ß x id)(A))) —> R, which is explained by Figure 58,
restricts to the identity on {e < u < a + Se}, and it maps the black region in
{a + Se < u < a + 9e } to the black region in its image. Moreover,
We
finally
wrap the set
into Z. The smooth
area
local
y(u, v)
for
m
G
]e + i(a + Se),
e
+
(i
+
The map y is constructed in the
Since y is
an
embedding
on
—
y(u
—
i(a
+
Se), v)
l)(a + Se)] and i
same
each part
way
as
=
1, 2, 3,...,
the map in
Step
see
Figure
58.
4 of Section 2.2.
167
y
2â
,
a
+
a
â
a
ele
Figure
{(m, v)
i
=
g
O, 1, 2,
p
(cp ((ß
...,
57: The left part of the
x
id)(A))) |
cf.
symplectic embedding,
composition
a
$
is
a
:=
x
a,
ocp
o
4. Verification of the estimate
We have to show that for any
intersection
O
we
To this end,
x
< e
id)(A)).
+
(i + l)(a + 8e)},
the map
-*
Figure
(ß
x
R4
59.
We conclude that the
id)
(7.9) in Proposition 7.5
point (xo, Jo)
R2
c(<P(A)nE(xo>yo))
holds true.
((ß

cp
of A into Z.
symplectic embedding
Step
((ß
Figure
(y
image
i(a + Se)
+
e
of its domain, and since â
y xid: cp
is
+ 9e
a
we
<&(A)CiE(XQtyQ)
may
assume
is empty.
the estimate
<
2e
(7.12)
g
]0, 1[ since otherwise the
that xq
Moreover, by
construction of the
embedding
have
0(m
+
2e,
v, x,
y)
=
0(m,
v, x,
y)
+
(0, 0, 0, â)
(7.13)
168
7. Proof of Theorem 4
l-~
2e3e
e
a
+ 8e
n
a
+ Se
n
Y
v
iL
1
1-6
58: The
Figure
for all
(m,
v, x,
that yo + ià
G
y)
e
to
map y.
A. It follows that for each yo
]â, 2a]
G
C
Case A. Assume first yo
:=
=
{0(m,
{ (m,
v, x,
v, x,
G
]â,â
y) \ 3e
y) |
e
<
]0, 1[
:=
TL such
]a, 2a]. In order
x
+
a[. Then E(XQtyo) intersects the "floor"
u
<
Ae
<
2e, 0
y) \
2ie
JO+I-â)),
Figure 59. We may therefore assume that (xo, Jo)
verify the estimate (7.12) we distinguish two cases.
F2
R there exists i
and
p(®(A)nE(xo,yo))
cf.
folding
{ 0(m,
v, x,
< u
<
(2i
+
\)e },
i
=
1,2,
y
<
2à }
169
*-
e
cf.
E(xQ,yo)
59. The set F
Figure
a
+ 9e
M
n
59: The floor F2 and the stairs Si and S2.
Figure
only,
2e
:=
F2
H
Foo.yo)
has
e(\
area
consists of the black thickened "arc" A drawn in
branch Bi
Bi(xo, yo), which for (xo, yo)
=
Indeed, since the
preimage
off from definition
Bi(x0,yo)
where ui
ui
=
(yo)
Si
=
(5, â 4- |)
e). The
Figure
set
Si
n
60 and of the
looks like in
under <£> is contained in
{0
<
<
y
Figure 60.
a}, we read
(7.11) that
{(",
=
of
—
v
+
f(u)x0,x0,yo) |
is defined
through
the
f(s)ds
mi
< m
< a
+
8e, 0
<
u
<
<5}
(7.14)
equation
=
(7.15)
jo-a.
J3e
Similarly,
(j, â
+
I)
{â
<
j
in
the branch
B2(xo, Jo)
=
#2(^0, Jo)
=
S2
n
E(xo<yo)
Figure 60. Indeed, since the preimage
â + a), we read off from (7.11) that
as
<
B2
H
in
{m
>
3e]
=
{ (m,
u
+
/(m)xo,
x0,
Jo) I
of
3e
looks for
(xo, jo)
=
S2 under $ is contained
< u
<
m2, 0
<
v
<
8
}
(7.16)
170
7. Proof of Theorem 4
M
Figure
where M2
U2
is defined
he
(7.15)
from
Subtracting (7.17)
0
â
<
a
—
the
using
/
=
n
(x0, Jo)
(j, â
=
+
§).
equation
f(s)ds
and
for
F(x0iVo)
through
+ Se
a
Ml
60: The set <&(A) n
«2(jo)
=
3e
2e
e
yo-â.
=
that
f(s)
f(s)ds
(7.17)
<
<
1 for all
Ml
s e
R
we
(7.18)
M2.
—
obtain
Ju2
Let p, be the
of
O(tA)
Pi
of the set
area
F(XO:Vo)
<$>(A)
next embed
we
E(Xo<yo). In order to
<Ï>(A) n E(XOtyo) into a
Pi
estimate the
set whose
capacity
capacity is
known.
Lemma 7.7 There exists
a
symplectomorphism
cp(p(<S>(A)nEixo,yo)))
Proof.
The set p
(<£>(<A)
in view of the estimate
curve
in
R2,
such that p
ment
given
c
(<&(A)
n
of U is
E(_XOtyo))
a
ofR2
such that
D(p + e2Y
R2 is diffeomorphic
(7.18) its boundary is
Figure
boundary
in the
E(XOtyo))
a
60. We therefore find
cf.
such that the
n
C
cp
smoothly
C U and
to
open disc, and
piecewise smooth embedded closed
simply connected domain U CR2
a
embedded closed
area(U)
proof of Lemma 7.6 (iii)
to an
U,
=
we
p +
find
curve
in
e2. Applying
a
R2,
and
the argu¬
symplectomorphism cp
171
of
R2
D(p
such that
e2),
+
as
cp(U)
D(p +
=
e2).
(p (<&(A)
Then cp
c on
monotonicity and the conformality
R2, Lemma 7.7 implies
=
=
It remains to estimate the number p
Q
=
{2e
(A)nE(x0tyo))
identities
cp(U)
C
desired.
In view of the
capacity
E(xo<yQ)))
fi
<
v
(7.16)
1
symplectic
c(cp(p(<S>(A)nEixo,yo))))
(/* e2).
(7.19)
+
=
(O(A)
area
n
E(Xo>yo)).
e}. Using the definitions
—
we can
A)
U
<
of the normalized
of
We abbreviate
ß
and y and the
estimate
area(p(B2)U p(Bi)U y~l(p(A))\
=
<
area({ (u,
=
area({(u,v) \(u,v)
=
area(ß(Q))
f(u)x0) \ (u, v)
+
v
e
ß(Q)})
ß(Q)})
area(Q)
=
G
=
e(\-e).
Therefore,
p
=
In view of
area
(7.19)
Case B. Assume
(0(«A)
we
E(XQ<yo))
n
conclude that
now
G
jo
[â
c
+
(B2
U
area
<
e(l-e) + (l-e).
(®(A)
SI
2a]. Then
+ a,
(F)
=
area
E(Xo>yo))
E(XQtyo)
<
5i
A)
U
2e.
intersects the stairs
S2 only,
and
$>(A)
where m/
E(XOty0)
n
=
=
U[(yo) and
{(u,
ur
v
+
f(u)x0, xo,yo) \ui
ur(yo)
lie in [3e,
(a 4- a)
and
=
a
+
(A)
is
n
F(^0i),0))
piecewise
smooth.
R2 is diffeomorphic
Moreover,
C
area(O(tA)nF(X0iy0))
=
(ur
-
u{) 8
<
to an
(a + 5e)
open disc, and its
—
<
e.
172
7. Proof of Theorem 4
Arguing
above,
as
of the estimate
Step
we
find
:
Z4(a)
a
In view
(7.9).
<—>-
E(x^yo))
proof of Theorem
We have constructed
^
n
2e. This finishes the verification
<
(7.12) and hence of the estimate (7.9) in Proposition 7.5.
5. End of the
timate
($(«A)
c
7.1
symplectic embedding $: i
of Proposition 7.5 there exists
Z4(7r) satisfying
the estimate
arbitrary, we conclude that fc (Z4(a))
implies that Theorem 7.1 holds true.
—
a
Since
(7.10).
0 for all
satisfying the es¬
symplectic embedding
a G
Z
^
]0,
a
n
and
< n
e
>
0
were
[. Proposition 7.4
now
D
Remarks 7.8
(B2n(n)) for any symplectic capacity
c(Sl). It would
proof suggest that £c (B2n(n))
1. We don't know
[23] and its
have
a
Lemma 1.2 in
c.
>
be
interesting
there exists
equivariance (7.13)
a constant
|d>(z)
Combining
L
of the
symplectic embedding
O
:
A
t->
4>(z')| >L\z-z'\
for all z,
z'e
A
the methods used in this section with Theorem C.6 in
following generalization
of
Corollary
7.2
Appendix
C
one
(i).
ofR2n such that there exist numbers a < n and A <
and a symplectomorphism ofR2n which embeds S into the truncated cylinder
Assume that S is
a
subset
D(a)
£C(S)
=
Ofor
We don't know t,c
also don't know
(Z2n(a))
3. The invariants
symplectic
x
R2"-3 x]-A,A[
any normalized extrinsic
anything
£c
Z
0 such that
>
-
therefore prove the
Then
to
proof of this.
2. In view of the
can
£c
for any
about Çc
Z2n(a).
C
symplectic capacity
symplectic capacity
(B2n(n))
nor
Çc
c
as
studied in
[31].
c on
and any
(Z2n(n))
and t,c considered in this section
intersection invariants
oo
are
if
a
g
c(Sx)
R2.
]0, n[. We
=
0.
special constrained
Appendix
A
Proof of
The purpose of this
we
1
Proposition
appendix
is to prove
1 of the introduction which
Proposition
restate as
Theorem A.l
dimensional
ume
Consider
manifold
open subset U
an
M endowed with
preserving embedding
cp: U
Assume first that cp: U
Proof.
a
if and only if
M
°->-
M is
^
of Rn and
smooth connected
a
volume form Q. Then there exists
a
Vol
(17, fio)
smooth
embedding
[
<
Vol
<
a
n-
vol¬
(M, £2).
such that
cp*Çl
=
£2o- Then
Vol(C/, fi0)
Assume
now
f
=
Ju
that Vol
embedding cp;U
<^->-
S2o
f
=
Ju
(U, fio)
<
(p*tt
fi
(
(M, fi). We
cp*Q,
fi
=
Vol(M, fi).
Jm
J<p(U)
Vol
M such that
=
are
going
to construct a
smooth
fio.
=
We orient Rn in the natural way. The orientations of R" and M orient each
open subset of Rn and M. We abbreviate the
a
measurable subset V of R"
Moreover,
we
Proposition
exists
Proof.
a
denote
embedding
We choose
an
a :
C
a
V2
(J^Li
Vk
Let en
:
V. To fix the ideas,
V2
c-^-
Rn be
a
r
(V, fio) of
centered at the
non-empty open subset ofRn.
|R" \ <r(V)|
=
origin.
Then there
0.
sequence
C
•
•
•
C
Vk
C
Vit+i
of non-empty open subsets of V such that Vk
=
Vol
write V for the closure of V in R".
VhR" such that
increasing
Vl
we
measure
the open ball in R" of radius
by Br
A.2 Assume that V is
smooth
and
by | V\,
Lebesgue
we assume
smooth
•
•
Vk+i
•
fork
—
1,2,... and
that the Vk have smooth boundaries.
embedding
IBi WVi)|
C
c
<
such that
2"1.
ai(Vi)
C
5i and
174
A. Proof of
Since
Vi
V2 and eri(Vi)
C
R" such that
a2\vl
=
Bi
c
and
a\Wi
c
B2,
02^2)
we
C
<*k\vk-i
induction
find smooth
we
o-jt-ilvt-i andak(Vk)
=
C
Bk
—
1,2,
smooth
The map
embedding
the estimates
a :
V
->
embedding
0-2 :
V3
^->-
2~2.
<
embeddings
ak
:
Vk+i
R" such that
<—>-
2~k,
<
R" defined
(A.1)
by al^
=
ak\yk
of V into R". Moreover, the inclusions
is
a
ak(Vk)
well defined
C
a(V) and
(A.l) imply that
\Bk\a(V)\
and
smooth
and
\Bk\ak(Vk)\
k
a
1
B2 and
\B2\cr2(V2)\
Arguing by
find
Proposition
<
\Bk\ak(Vk)\
<
2~k,
so
\Rn\a(V)\
This
completes
Our next
goal
the
=
lim
\Bk\a(V)\
=
0.
proof of Proposition A.2.
is to construct
embedding of R" into the connected ndimensional manifold M such that the complement of the image has measure zero.
If M is compact, such an embedding has been obtained by Ozols [29] and Katok
[19, Proposition 1.3]. While Ozols combines an engulfing method with tools from
Riemannian geometry, Katok successively exhausts a smooth triangulation of M.
Both approaches can be generalized to the case of an arbitrary connected manifold
M, and
we
a
smooth
shall follow Ozols.
We_abbreviate
R>0
=
{r
G
R |
r
>
0} and R>0
=
R>o
U
{00}.
We
endow
R>o with the topology whose base of open sets consists of the intervals
]a, b[ C R>o and the subsets of the form ]a, 00]
]a, oo[ U {00}. We denote the
=
Euclidean
norm on
Proposition
be
a
R"
by ||
•
|| and the unit sphere in R" by Si.
A.3 Endow Rn with its standard smooth structure, let p: Si
continuous function and let
S
=
\x
gR"
0
<
||x||

R>o
175
Remark A.4 The
such that the rays
diffeomorphism guaranteed by Proposition A.3
emanating from the origin are preserved.
Proof of Proposition A3. If p(S{)
that p is bounded,
Proposition
{oo} nor
p(Si)
that neither
A.3 has been
Using
addition that ro
<
his notation, the
1 and that 6]
<
is
nothing
proved by
to
prove. In the
case
[29]. In the
case
Ozols
bounded, Ozols's proof readily extends
p is
=
this situation.
{oo}, there
=
may be chosen
only
modifications needed
2, and define continuous
to
Require in
functions ßi : Si —>
are:
R>oby
ßi
With these minor
In the
to
adaptations
following
the
shall
we
min
i,
p
e,; +
-
§1.
proof in [29] applies
use
some
word
by
word.
basic Riemannian geometry.
[20] for basic notions and results in Riemannian geometry.
dimensional
point
N
p
G
Corollary
an
=
complete
by C(p).
Riemannian manifold
We refer
Consider
(N, g). We denote the
an n-
cut locus of a
A.5 The maximal normal
n-dimensional
neighbourhood N \ C(p) of any point p in
complete Riemannian manifold (N, g) is diffeomorphic to Rn
endowed with its standard smooth structure.
identify the tangent space (TpN, g(p)) with Euclidean
space R" by a (linear) isometry. Let exp„ : R" -> N be the exponential map at
p with respect to g, and let Si be the unit sphere in Rn. We define the function
p, : Si -* R>0 by
Fix p
Proof.
G
N. We
p(x)
inf{t
=
Since the Riemannian metric g is
Theorem
7.3]. Let S
7.4
|
expp(fx)
complete,
G
C(p)}.
(A.2)
the function p is continuous
[20, VIII,
C R" be the starlike domain associated with p. In view of
diffeomorphic to R", and in view of [20, VIII, Theo¬
N\C(p). Therefore, N\C(p) is diffeomorphic
haveexp„(S)
A.3 the set S is
Proposition
rem
0
>
(3)]
we
=
toR".
A main
cases
of
a
ingredient
of
our
proof of Theorem
theorem of Greene and Shiohama
Proposition A.6 (i) Assume that fii
set U ofRn such that Vol (U, fii)
phism ifr of U
such that
i/r*fii
=
is
=
fio-
a
A. 1
are
following
two
special
[11].
volume form
\U\
the
<
oo.
on
the connected open sub¬
Then there exists
a
diffeomor¬
176
A. Proof of
(ii)
Assume that fii is
Then there exists
End of the
a
a
volume
we can assume
(M, fi) be
that
\U\
(M, fi). We
on
closed submanifold in
as a
N.
Riemannian metric. A direct and
Riemannian metric is
on
point
R"
p
G
given
in
measure
we see
proof
to
:
=
Let
Ui, Ü2,
(AT, fi).
Rm. We
Corollary
—>
R>o
as
Si}
G
(C(p))
(N \ C(p), fi)
a
(A.3)
can
then take the induced
existence of
A.5
a
complete
in
identify (TpN, g(p))
(A.2). Using polar coordi¬
we
C
Rn
also has
measure zero
In
a
->
=
Vol
(see [3, VI,
(AT, fi).
(A.4)
diffeomorphism
N\C(p).
Rw, if necessary, we can
view of (A.3) and (A.4) we then have
reflection of
\U\
<
Si
A.5 there exists
8 with
preserving.
Case 1. \U\
dM. Then
\
It follows that
Corollary
composing
of
exp
8: Rn
orientation
Vol
some
{p(x)x |x
=
C(p)
so
Vol
After
=
M
=
from Fubini's Theorem that the set
zero, and
Corollary 1.14]).
According
set N
if necessary,
[28].
N. As in the
C(p)
has
fio.
enlarging U,
elementary proof of the
with R" and define the function p
nates
=
oo.
boundary, there exists a complete Rie¬
Indeed, according to a theorem of Whitney [37], N can
be embedded
a
T/r*fii
=
connected manifold without
a
mannian metric g
Fix
such that
in Theorem A.l. After
Vol(M, fi)
=
Rn such that Vol (R", fii)
1
A.l
as
Vol
=
\U\
Since N is
on
diffeomorphism \/f ofRn
proof of Theorem
Let U C R" and
form
Proposition
=
assume
that 8 is
Vol(R",<5*fi).
(A.5)
oo.
be the
countably
for each i. Given numbers
a
many components of U. Then 0
and b with
-oo
<
a
<
è
<
oo we
"open strip"
Sal
We
can
therefore
inductively
define ao
=
—
oo
Vol(Sa,_l!Û(,S*fi)
Abbreviating S;
For each i
Si.
>
=
1
Sai_ua,
we
choose
In view of
that R"
A.2
volume,
Vol
(£/,-, o-*T*<5*fi)
In view of
we can now
Proposition
oo,
oo] by
U(>1
S(-.
preserving diffeomorphism t, : R" ->
a smooth embedding <r,- : Ui '—> R" such
After composing er, with a reflection of R", if
orientation preserving. Using the definition of
we
zero.
conclude that
Vol
=
—
find
that a; is
we can assume
the
]
\Ut\.
=
=
g
orientation
an
Proposition
\ ai(Ui) has measure
necessary,
then have R"
we
and a,-
(a,-(£/«), T*«5*fi)
A.6
(i)
=
Vol(R", r*<5*fi)
therefore find
we
=
Vol(S(-, 5*fi)
diffeomorphism ^
a
of
=
Ui such
that
f*
We define cpi
Ui
:
°->-
(a*T*8*Q)
M to be the
fi0.
=
composition
of
(A.6)
diffeomorphisms
and smooth
embeddings
%
Ui
The
Ui % Rn \ Si cRn
that
identity (A.6) implies

M to be the
identity (A.7) implies
plete.
that
=
oo.
Proposition
A.6
(ii)
shows that
of R" such that
diffeomorphism \Js
U czRn
The
C
fio. The smooth embedding
l/r*5*fi
We define cp
N\ C(p)
oo.
(A.5)
a
—
=
<p*S2
-^
=
fi0.
composition
^
Rn
<p*fi
=
^
(A.7)
of inclusions and
N\ C(p)
c
diffeomorphisms
M.
fio. The proof of Theorem A.l is
com¬
\Ut\.
178
A. Proof of
Proposition
1
Symplectic capacities
B
and the invariants cB and
cc
In the first two sections of this
which
symplectic capacities
appendix
are
we
collect and prove those results
on
relevant for this thesis. We refer to [2, 6, 7, 14,
15, 17, 18, 21, 26, 32, 33, 35] for
information
symplectic capacities. In
we
Chapters 1 to 6 with
the
In
last
sections
two
we
symplectic capacities.
compare c# with the symplectic
diameter and interpret ce as a symplectic projection invariant.
Section B.3
B.l
more
Intrinsic and extrinsic
We say that
on
compare the invariants eg and cc studied in
a
symplectic capacities
R2" symplectically
subset S C
R2"
on
embeds into F
R2n
c
if there
symplectic embedding cp of an open neighbourhood of S into R2" such
cp(S) C F. We again write Z2n(a) for the symplectic cylinder D(a) x R2"-2.
exists
that
a
Recall the
Definition B.l An intrinsic
with each subset S of
ating
following
axioms
Monotonicity: c(S)
A2.
Conformality: c(kS)
A3.
Nontriviality:
0
<
<
For any subset S of
if S
c(T)
=
X2c(S)
c(B2n(n))
symplectic capacity
A3'. Normalization:
cal
on
G
(R2",
coo) is
a
in such
[0, oo]
map
a
c
associ¬
way that the
satisfied.
are
AI.
An intrinsic
symplectic capacity
R2" a number c(S)
R2"
we
=
for all k
and
R2n
c on
c(B2n(n))
symplectically
g
R
embeds into F.
\ {0}.
c(Z2n(n))
< oo.
is normalized if
c(Z2n(n))
=
n.
define the Gromov width
wc(S) and
the
capacity z(S) by
wq(S)
:—
sup{a \
z(S)
:=
inf
{a \
B2n(a) symplectically embeds into S],
S symplectically embeds into Z2n(a)}.
cylindri¬
180
B.
It follows from Gromov's
and the invariants cq and cc
Symplectic capacities
Nonsqueezing
Theorem stated in
Example 1 of the in¬
troduction that both wg and z are normalized intrinsic symplectic capacities on
R2". Another example of a normalized intrinsic symplectic capacity on R2n is the
Hofer-Zehnder
Proposition
capacity
capacity [18].
B.2 For any subset S
c on
R2n
ofR2n
we
wG(S)
B2n(a)
If
Proof.
S is
°->-
<
c(S)
symplectic embedding,
a
supremum,
we
find
embedding,
we
conclude from the axioms
infimum
find
u>g(S)
c(S)
Remark B.3 If S is
symplectic
<
z(S).
<
<
conclude from
mono¬
a
<
=
c(S). Similarly, if S
—
z(S).
open connected subset of
an
we
B2n(a)
c(S). Taking the
<—> Z2n(a) is a symplectic
that c(S) < c(Z2n(a))
a. Taking the
and normalization that
tonicity, conformality
we
and any normalized intrinsic
have
sition B.2
R2,
If
the
inequalities
in
Propo¬
are equalities, see Corollary B.9(i).
n
starshaped domains S C R2" with arbitrarily small Gromov width and
1 (see [15]). Still, for an ellipsoid and, more generally, a convex Rein¬
z(S)
hardt domain all normalized symplectic capacities on R2" coincide [15]. It is
>
2, however, there exist
bounded
=
conjectured that
As in
(R2",
co0).
this holds for any bounded
Chapter
1
we
denote
domain [36].
the group of
O
symplectomorphisms
of
We also recall the
Definition B.4 An extrinsic
with each subset S of
ating
following
by £>(n)
convex
axioms
are
satisfied.
AI.
Monotonicity: c(S)
A2.
Conformality: c(XS)
A3.
Nontriviality: 0
An extrinsic
symplectic capacity on (R2n, coo) is a map c associ¬
R2" a number c(S) G [0, oo] in such a way that the
<
<
=
X2c(S)
c(B2n(n))
symplectic capacity
A3'. Normalization:
if there exists cp
c(T)
and
c on
c(B2n(n))
=
for all A.
R2n
G
g
R
£>(n) such that cp(S)
\ {0}.
c(Z2n(n))
<
is normalized if
c(Z2n(n))
=
n.
oo.
C F.
B.l. Intrinsic and extrinsic
symplectic capacities
on
R2"
181
Comparing Definitions B.l and B.4 we see that any intrinsic symplectic capacity
on R2n is an extrinsic symplectic capacity on R2". The converse is not true as we
shall
in
Proposition
B.6.
For any subset S of
R2"
see
wG(S)
:=
z(S)
:=
It follows
define
we
there exists cp
sup{a |
inf{a | there exists
cp
g
<£>(«) such that
cp
(B2n(a) )
G
D(n) such that
cp
(S)
C
S},
C
Z2n(a)].
Nonsqueezing Theorem that both wq and z are
symplectic capacities on R2w. Other examples of normal¬
symplectic capacities on R2" are the first Ekeland-Hofer capacity
again
from Gromov's
normalized extrinsic
ized extrinsic
[6] and used in the proof of Theorem 1.4, and the
ci, which was introduced in
displacement
e
introduced in
c on
R2n
wg(S)
=
bounded components
The
Proof.
and any normalized extrinsic symplectic
have
we
Moreover, wg(S)
[17].
ofR2n
B.5 For any subset S
Proposition
capacity
energy
c(S)
c(S), andz(S)
ofR2n \
=
z(S).
<
z(S) where
S is the union
of S
with the
S.
inequalities wg(S)
c(S)
<
<
z(S)
are
proved
in the
same
way
as
B.2.
Proposition
The inequality wg(S)
fix
we
to
e
>
0 and
Proposition
<
that cp
assume
is obvious. In order to prove
u>g(S)
O
In view of the
z(S)
z(S)
<
ZZn(a).
monotonicity
we assume
<p(S)
C
z($)
that cp
Moreover, if U is
bounded component of
that
c
—
Z2n(a),
<
z(S).
S, and
of
g
>
a.
z
we
proof
wg(S)
Taking
have
>
and
B2n(a)
^>|ß2«(ö_e)
a
—
z(S)
<
z(S).
embeds S into
so
e.
cp(U)
C
R2" \ S,
Z2n(a).
a.
<p\B^(a-e)0
was
we
find
>
e
over a
In order to prove
Z2n(a).
Taking the infimum
Proposition B.5 is complete.
g(S)
>
According
=
Since
the supremum
bounded component of
z(S)
of
so
D(n)
R2" \ cp(S),
and hence
The
a
£>(«) such that
g
particular,
(B2n(a e))
arbitrary, we conclude that wg(S)
Ûg(S) > wG(S).
In
embeds
symplectically
C.l there exists O
wg(S)
into S.
Then
then
cp(S)
cp(U)
is
c
a
It follows that
over a we
conclude
proposition shows that neither the first Ekeland-Hofer capacity ci
displacement energy e nor the outer cylindrical capacity z are intrinsic
symplectic capacities on R2n. Let S1
{(«, v) \ u2 + v2 1} be the unit circle.
The next
nor
the
=
=
182
B.
B.6 For the standard
Proposition
we
Symplectic capacities
Lagrangian
and the invariants eg and cc
torus
Tn
S1
=
x
•
•
S1
x
in
R2n
have
ci(Tn)
while
c(Tn)
=
Ofor
According
Proof.
=
any intrinsic
Théorème
to
e(Tn)
(b)
on
(i) in [17] implies ci(F")
z(Tn). Since clearly z(Tn) <
<
Tt
and
so
that
c
B.7
ci(Tn)
=
page 43 of
c on
[33]
R2n.
we
have
ci(F")
is
an
n,
n, we conclude that
e(Tn)
z(Tn)
<
<
n
we assume
symplectic capacity on R2". Fix e > 0. In view of Theorem
a symplectic embedding cp : S1 °-> D(e). The product
intrinsic
below there exists
x
cp
symplectically
x
cp: Tn
embeds F" into
c(Tn)
e
=
e(Tn), and Proposition B.5 shows that
<
the first assertion follows. In order to prove the second assertion
(ii)
Since
<
n
=
symplectic capacity
Theorem 1.6
e(Tn)
z(Tn)
=
>
0
was
arbitrary
<
D(e)
^
Z2n(e),
x
and
c(z2n(e))
and since
c
•
•
•
x
D(e)
Z2n(e)
C
so
=
-c{z2n(n)).
(Z2n(n))
<
oo
it follows that
c(Tn)
=
0
as
claimed.
Symplectic capacities
B.2
If
>
n
2 the
problem.
Since
of
computation
In this section
B2(n)
plectic capacity
=
c on
we
Z2(n)
R2
a
on
R2
capacity
of
a
subset of
show that for subsets of
we can assume
that any
is normalized. Given
and S its interior and its closure and
a
of S is denoted
by p(S),
and if S is
(i) wG(S)
=
a
subset S
swpiP(Si).
usually
a
difficult
the situation is different.
(intrinsic
subset S of
R2
or
we
extrinsic)
sym¬
denote
Int S
by
R2 \ S. The outer Lebesgue mea¬
measurable, p(S) denotes its Lebesgue
measure.
Theorem B.7 Consider
is
the components of Int S. As before, S is
by S(
the union of S with the bounded components of
sure
R2
R2w
ofR2.
B.2.
Symplectic capacities
(ii) z(S)
(iii) If S
Proof, (i)
bounded, z(S)
We abbreviate
S, then cp(D(a))
Then there exists
the introduction
u>g(S)
>
s
e.
—
=
cp
i, and
some
that
S(- such that p(Si)
find
we
Since
e
a
g
is
p(S)
oo.
=
symplectically embeds D(a) into
wq(S) < s. In order to prove the
so
s G
]0, oo]. If s
e
]0, oo[
fix
we
e
g
]0, s[.
According
Proposition 1
e) into S,-, and
symplectic embedding of D(s
conclude
that
was
we
]0, s[
u>g(S) > s.
arbitrary
>
s
—
to
e.
—
wq(S)
=
if
oo
embeds S into
symplectically
unbounded, z(S)
sup,- p (Si). If cp
we can assume
similar argument shows that
(ii) If
p(S). If S
:=
s
S,- for
c
inequality
reverse
183
p(S).
=
is
R2
on
=
P((P(S))
s
=
oo.
Assertion
of
so
A
(i) thus follows.
D(a), then
<
ß(D(a))
=
a.
Taking the infimum over a we find that JZ(S) < z(S). In order to prove the reverse
inequality we can assume that ~ß(S) < oo. Fix e > 0. In view of the definition
of p(S) there exists an open neighbourhood U of S such that p(U) < ~ß(S) + e.
By Proposition 1 we find a symplectic embedding cp of U into D (p(S) + e).
Therefore z(S) < p(S) + e. Since e > 0 was arbitrary we conclude that z(S) <
~ß(S). Assertion (ii) thus follows.
(iii) Using Proposition B.5,
we
the
identity in (ii)
and the fact that S is measurable
find
z(S)
In order to prove
=
z(S)
>
z(S)
=
p(S)
=
p(S).
p(S) we therefore need to show z(S) < p(S). By con¬
struction, S is compact and R2 \ S is connected. Therefore, each path-component
of S is
simply
z(S)
<
connected. Since S is compact,
Lemma B.8 There exists
thatp(W)
<
in
simply
connected open
is finite. Fix
e
>
neighbourhood
0.
W
of S
such
+ 2e.
p(S)
We say that
Proof.
a
p(S)
a
subset of
R2. By
R of R2
R2 is elementary
if it is the union of
finitely
definition of p and since S is compact
many
find
an
rectangles
<
such that S C R and p(R)
p(S) + e. Let U be
elementary subset
one of the finitely many components of R. The elementary set U is diffeomorphic
open
to
a
an
open disc from which k closed discs have been removed. Since A n U is
compact subset of U all of whose path-components
set U
we
\
S is open and connected.
simply connected, the
curves y, in U \ A and
are
We therefore find k
184
B.
Symplectic capacities
and the invariants c# and cc
elementary neighbourhoods Ni of y, such that the elementary subset V := U \
(J N{ is a simply connected neighbourhood of U n S. Applying this construction
to each component of R we obtain disjoint simply connected elementary subsets
1 curves yl in
Vi,..., Vi of R whose union contains <8. We finally choose /
R2 \ U W anc* elementary neighbourhoods Nl of y' such that the elementary set
—
l-i
/
W
(J Vi
:=
(J Nl
U
i=l
is
simply connected
neighbourhood of J
P(W)
<
p
and such that p
i=l
((J Nl)
< e.
Then W is
a
simply
connected
and
((J V;-)
+ p
([J JV'")
<
p(R)
+
e
<
p(4)
+ 2e.
Lemma B.8 thus follows.
Let W be
simply
neighbourhood of S as guaranteed by
(p(W)). According to Proposition
diffeomorphic
A.6 (i) we therefore find a symplectomorphism cp : W —> D (p(W)). Since cp(S)
is a compact subset of D (p(W)), we find a g ]0, p(W)[ such that cp(S) C D(a).
According to Proposition C.l we find a symplectomorphism O of R2 such that
<P_1|z)(a)- The map $_1 is then a symplectomorphism of R2 which
^>\rj(a)
a
connected open
Lemma B.8. Then W is
to D
=
embeds S into
was
arbitrary
D(a). It follows that z(S)
we
conclude that z(S)
claim in assertion
<
< a
<
p(W)
<
/x(S)
+ 2e. Since
e
>
0
p(S). This completes the proof of the first
(iii).
unbounded, then cp(S) is unbounded for any diffeomorphism cp of R2,
oo. Assertion (iii) thus follows, and so the proof of Theorem B.7
z(S)
If S is
and
is
so
—
complete.
Corollary
and p
B.9
(Int S)
capacity c on
c(S)
p(S).
(i) Assume that S is
~ß(S). Then c(S)
R2. If in addition
=
=
p
a
subset
of R2
such that Int S is connected
~ß(S)for any normalized intrinsic symplectic
(Int S) is finite, then S is measurable and so
=
(ii) Assume that S is
pi
(Int S)
extrinsic
=
a
bounded subset
ofR2
such that Int S is connected and
p(S). Then S is measurable and c(S)
symplectic capacity
c on
R2.
=
p(S) for
any
normalized
B.2.
Symplectic capacities
Let
Proof.
c
and
wG(S)
so
c(S)
(ii)
S c 4
=
If /x
(Int S)
monotonicity
fore S
symplectic capacity on R2.
Theorem B.7 and the assumptions on S
in
(ii)
c(S)
<
p(S).
=
p
the
185
be any normalized intrinsic
Proposition B.2, (i)
and
R2
on
\
JÎ(S)
=
p(S)
=
Int S is
{p(U) \
inf
=
additivity
of p
measurable, and
Since S is bounded, p
(Int S)
=
S C U,
(S \ Int S) <p(<8\ Int S)
normalized extrinsic
(iii)
c(S)
so
wG(S)
=
U
open},
pZ(S \
—
Int
S)
=
0. There¬
p(S).
p(S) is finite.
=
/x(4)
p(S)
R2.
on
symplectic capacity
in Theorem B.7 and the
wG(S)
so
that
imply
It follows that S is measurable and that
and
have
This and the inclusion
yield
Tx
and
(Int S)
p
=
we
(Int S) is finite, the identities
of p and the
Int S U S
z(S)
<
In view of
c(S)
=
=
wG(S)
p(lnt S)
<
—
assumptions
c(S)
p(S)
<
as
z(S)
p
=
=
on
/x
-
(Int S)
(Int S).
In view of
S
p(S)
we
=
embeds into
0.
Let
now c
be any
Proposition B.5, (i)
have
/x(IntS)
=
wG(S)
claimed.
We recall that Gromov's
plectically
=
Nonsqueezing Theorem states
a cylinder Z2n(A) only if A > a.
that
a
ball
B2n (a)
sym¬
Corollary B.10 Fix a G ]0,n] and a normalized extrinsic symplectic capacity
R2. Then the Nonsqueezing Theorem is equivalent to the identity
c
on
inf
where cp varies
Z2n(n),
over
all
and where p:
c(p(cp(B2n(a)ty=
symplectomorphisms o/R2"
Z2n(n)
—>
B2(n)
is the
(B.l)
a
which embed
B2n(a)
into
projection.
Assume first that the
Nonsqueezing Theorem does not hold. Then there
exist a' > 0 and A < a' and a symplectic embedding cp: B2n(a!) ^-> Z2n(A).
According to Proposition C.l we find a" e ]A, a'[ and a symplectomorphism <£>
Proof.
of
R2"
such that
<S>(B2n(a")\
C
Z2n(A).
186
B.
f;A
Notice that
<
composition a-1
o
into the subset Z2n
a
<
$
o a
is
a
of
symplectomorphism
Z2n(n),
The axioms for the normalized
-%A
< a we
C
symplectic capacity
Assume
we
that
now
then find
a
conclude that
(B.l) does
c
We abbreviate U
B.9
:—
(i), Proposition
z(U)
=
We therefore find
(\p-
x
id2n-i)
squeezing
(B.l) does
R2"
D
(-â)
°

2e.
-
(B.2)
Since U is open and connected,
B.5 and the estimate (B.2)
p(U)
R2n by
which embeds
(p (B2n(a)))
c
and
(p (<p (B2n(a))\\
(cp (B2n(a))).
of
not hold.
not hold. Since
symplectomorphism
^-z
of
c(p((a-1oOo«)(ß2M(a))))
Since
m^
z
i.e.,
((a"1 o$oa) (Vn(a)))
p
and the invariants cb and cG
We denote the dilatation
n.
(Â)
Symplectic capacities
Corollary
imply
wG(U)
<
c(U)
<
a-2e.
e). The composition
symplectic embedding ijr: U °->- D(a
embeds
into
B2n(a)
Z2n(a e) and so the Nonsymplectically
a
—
—
Theorem does not hold.
0 for any intrinsic symplectic
According to Proposition B.6 we have c(S1)
n for the first Ekeland-Hofer capacity, the dis¬
capacity c on R2 and c(S1)
and
the
outer
placement energy
cylindrical capacity. The following result, which
=
—
was
pointed out to me by David Hermann,
symplectic capacities on R2.
shows that there exist other normalized
extrinsic
PropositionB.il (D. Hermann) For any
extrinsic symplectic capacity c on R2 such
Proof. Fix
a
G
[0, n]. For
r
c(rD(n))
>
0
:=
a
[0,n] there exists
e
c(Sl)
that
=
a.
we set
r2n
and
c(rS^)
:=
r2a,
a
normalized
B.3.
Comparison
and
given
an
of cb and cG with
arbitrary
R2
subset S C
c(S)
symplectic capacities
:=
we
187
set
max{u>G(S),or(S)}
where
wG(S)
:=
sup
| r2n
| there exists
cp
a(S)
:=
sup
I r2a
\ there exists
cp e
Then
is
c
c(Sl)
=
a
cc(S)
n
n
=
C
£>(2) such that
cp
\rSl\
S
C
symplectic capacity
on
|,
S
\
.
R2,
and
D
cB(S)
>
(rD(n))
a.
For any subset S of
For
cp
well-defined normalized extrinsic
Comparison
B.3
S)(2) such that
e
of c# and cc with
R2"
define the
we
symplectic capacities
invariants cb and cG
symplectic
by
:=
inf {a
\ S symplectically embeds into B2n (a)},
(B.3)
:=
inf {a
C2n (a)}.
(B.4)
1, the invariants
2, the invariants
\ S symplectically embeds into
cb
and cc coincide with the
Cß and cc fulfill all axioms of
cylindrical capacity
a
z.
For
normalized intrinsic sym¬
R2" except that they are infinite for the symplectic cylinder.
plectic capacity
Recall from Proposition B.2 that the cylindrical capacity z is the largest normal¬
ized intrinsic symplectic capacity on R2".
on
o/R2"
we
cc(S)
<
cB(S).
The claim follows from the inclusions
B2n(a)
Proposition
B.12 For any subset S
z(S)
Proof.
Remark B.13 If
equalities:
n
<
For
>
have
—
polydisc
cb(S)
nn.
< n
C
C2n(a)
2, the inequalities in Proposition B.12
cc(P). Moreover,
Cß(C2n(n))
we
a
n
<
have
P
we
=
see
P2n(n,... ,n,a)
from
(This example
cG(S).)
(1.5)
and
with
(1.9)
is extremal in the
a
>
that
sense
n
C
are
we
Z2n(a).
in
D
general
have
cG(C2n(n))
not
z(P)
=
that for any S C
n
=
<
R2n
188
B.
B.4
Comparison
of cB with the
Recall that for any subset S of
d(S)
Symplectifying the
ds(S) defined by
ds(S)
:=
For convenience
R2"
:=
sup{|z-z'|
\d(cp(S))
\
d(S)
\z,z'
we
:=
HS)
Proof.
ball
=
1, then S(S)
Fix S C
eS}.
symplectic
S into
invariant
R2" I.
invariant 8(S) defined
by
2
-^ I
(
n
by
(B.5)
2
ds(S).
Theorem B.14 For any subset S
and ifn
is defined
obtain the
symplectic
ds(_S)\
rather than with
d(S)
symplectically embeds
cp
shall work with the
8(S)
and the invariants Cß and cG
symplectic diameter
the diameter
Euclidean invariant
inf
we
Symplectic capacities
R2".
=
<
Cß(S)
o/R2"
we
cB(S)
<
=
have
T^-<5(S)
(B.6)
2n + 1
p(S).
The circumradius
R(S)
of S is the radius of the smallest
containing S, i.e.,
R(S)
:=
infÏR
I
there exists
where rw denotes the translation
invariant R(S)
RS(S)
we
:=
obtain the
z i-> z
|
cp
Notice that each translation rw of
(B.3)
such that
+ w of
symplectic
infÏR (cp(S))
of the invariant
R2"
w G
C
B2n
(xR2)}
R2". Symplectifying the Euclidean
invariant
symplectically
R2"
rw(S)
is
embeds S into
R2"
symplectic. Comparing
cb(S) with the definitions of R(S) and RS(S)
}
.
the definition
we
therefore
find that
cB(S)
=
n(Rs(S))2.
(B.7)
B.5. The invariant cG
The main
ingredient
symplectic projection
as a
of the
proof of Theorem
\d(T)
R(T)
<
2
valid for any subset T of
R2".
invariant
B. 14
are
189
the
inequalities
-^—d(T)
<
(B.8)
y 2n + 1
While the left
inequality
in
(B.8) is obvious, the
right inequality is the main content of Jung's Theorem [4, Chapter 2, Theorem
11.1.1]. Applying (B.8) to all symplectic images cp(S) of S in R2" and taking the
infimum
These
find that
we
inequalities,
Assume
now
the definition
R2.
S C
and the
(B.5)
The left
identity (B.7) imply (B.6).
in
inequality
(B.6) and Theorem
B.7
(ii) show
that
S(S)
In order to show that 8 (S)
R2
and denote the
convex
inequality [4, Chapter 2,
n
Since cp
cB(S)
=
>
<
cB(S)
~ß(S)
Theorem
>
z(S)
p(S).
=
that cp
symplectically embeds S into
cp(S) by conv^(S). Applying the Bieberbach
11.2.1] to <p(S) we find
we assume
hull of
(d{(p(S))\
=
p(convcp(S))
arbitrary it follows that S(S)
p(S). The proof of Theorem B.l4
was
>
is
]Z(cp(S))
>
=
p(S).
We conclude that
p(S).
S(S)
=
complete.
Remarks B.15
1. Assume that
2. It follows from the Bieberbach
inequality [4, Chapter 2,
inequality in (B.6)
(B2n(n))
(B2n(n))
whether
the
know, however,
right inequality in (B.6) is sharp.
n
>
Theorem 11.2.1] thatS
is
sharp.
We don't
2. The invariants 8
invariants for
B.5
As
symplectic
n
pointed
extent the
area
out in
(y)
and cb
domains"
The invariant cc
was
going
=
as a
as
=
so
n(Rs)2
the left
are
of "Euclidean
studied in [31].
[10, p. 580], symplectic capacities
point precise
examples
symplectic projection
of two dimensional
to make this
and
cB
=
on
symplectic projections
for the invariants
z
and cG.
invariant
R2"
measure
of
set.
a
We
to some
are now
190
B.
For i
=
1,...,
we
dxi
A
dyi
on
{zen\zj=0 forjî},
=
orthogonal projection R2"
denote the
F; is denoted
sm(S)
inf min
l<i<n

pi, and the outer
ßi. Given any subset S of R2"
where cp varies
and the invariants cb and cG
set
n we
Ei
and
Symplectic capacities
=
inf
max
<P
l<i<n
ßi (pi(cp(S))),
ßi (pi(cp(S)))
symplectic embeddings
of S into
R2".
Remarks B.l6
1. If S C
i
=
1,
...,
R2"
n, and
is
Lebesgue measurable,
=
sm(S)
Composing
with the linear
(Zl,
we
Z2,
=
inf
mm
<P
t<i<n
the sets
pi(cp(S))
C
F(-,
inf
max

(Zi,
S
(pi(cp(S))).
R2"
c-^
Z2,
•
•
•
,
in the definition of sm(S)
Zi-l, Zl, Zi+l,
.
.
.
,
Zn)
R2",
inf pi
=
(pi(cp(S))).
 0.
symplectic embedding cp: S °->- R2" such that
assume
According to Theorem B.7 (ii) we find a symplectic
embedding \(r of pi (cp(S)) into D (sm(S) + 2e). The composition (iff x id2n-2) °
cp symplectically embeds S into Z2n (sm(S) + 2e). Since e > 0 was arbitrary, we
conclude z(S)
The
sm(S).
<
<
sm(S) +
e.
sm(S). The
equality cG(S)
=
inequality sm(S) < z(S)
proved in the same way as
reverse
sm(S) is
is obvious.
the
equality z(S)
=
B.5. The invariant cG
as a
symplectic projection
Remark B.18 The invariants sm
tion invariants
as
studied in
[31].
=
z
and sm
=
invariant
cG
are
191
special symplectic projec¬
192
B.
Symplectic capacities
and the invariants Cß and cG
C
The Extension after Restriction
Recall that
subset U of R" is said to be starlike if U contains
a
for every
point
following
result is well known
Proposition
x
G
U the
straight
R2n
[6]
R2".
such that
OaU
The purpose of this
reproved
a
a
point
p such that
is contained in U. The
x
below.
R2n
c-^
is
a
symplectic embedding of
a
any subset A C U whose closure
Then
is contained in U there exists
o/R2"
$A
line between p and
and
C.l Assume that cp: U
bounded starlike domain U C
in
Principle
for
compactly supported symplectomorphism
<P\a-
=
appendix
is to prove
version of
a
Proposition
C. 1 for
un¬
bounded starlike domains.
symplectic embedding cp: U °->- R2" of a starlike do¬
main U C R2". We say that the pair (U, cp) has the extension property if for each
subset Act/ whose closure in R2" is contained in U there exists a symplecto¬
<P\amorphism <£>a of R2" such that <$>a\a
Definition C.2 Consider
a
=
The
following example
shows that not every
pair (U, cp)
as
above has the
ex¬
tension property.
Example
C.3 We let U C
methods used in
cp: U
c->
R2
Step
1 and
such that
Step
cp(k, 0)
any subset A of U
to a
be the
strip ]1, oo[
4 of Section 2.2
=
containing
diffeomorphism of R2.
Observe that if
R2
(£, 0),
the set
sense.
still
k
]
find
2, 3,
=
-
a
However, the map
1, 1[. Combining the
symplectic embedding
Then there does not exist
2, 3, ...} for which cp\^ extends
O
(U, cp) has the extension property, then
is bounded is bounded.
(U, cp)
=
{(k, 0) \
that each subset A c U whose closure in
cp(A)
k
we
x
R2"
is contained in U and whose
Example
following example is
The map cp in
cp in the
cp is proper in the
does not have the extension property.
sense
image
C.3 is not proper in this
proper in this sense, and
194
C. The Extension after Restriction
Example C.4
Let U C
R2
A
where
/
:
R
]0, ^[ is
—>-
strip
R
x
]
-
1, 0[, and let
{(x,y)eU\\y + l\<f(x)}
=
a
be the
Principle
smooth function such that
/ (\
J~R
-
f(x))dx
<
(C.l)
oo,
- R2 such that
cp(x, y)
cf.
Figure
(x, y) if x
>
1
and
61. In view of the estimate
contains the
such that
=
=
infinite volume, to
=
we
(—x, —y)
(C.l) the component
C of
find
if
x
a
<
symplectic
—1,
R2 \ cp(A)
which
Any symplectomorphism <£>a of R2
<p\a would map the "upper" component of R2 \ A, which has
O
C. This is impossible.
point (1,0)
<&a\a
cp(x, y)
2.2
has finite volume.
195
C.4 shows that the
Example
cannot be omitted.
also
impose
smooth
a
curve
mild
y
:
assumption (C.2)
For technical
convexity
[0, 1]
length(y)
On any domain U C R"
we
define
where the infimum is taken
=
z'.
Then
\z
-
z'\
exists
a
constant À
>
a
À
=
convex
1. It is not hard to
which is not
a
Proof.
Step
-
U
U
x
-»
R
=
z
by
y
R" is
C
:
G
a
[0, 1]
—
U with
y(0)
U.
Lipschitz
domain if there
-
a
z'\
for all z,
Lipschitz
z'
g
U.
domain with
Lipschitz
a
constant
are
starlike domain with smooth
not
Lip¬
boundary
:
U
°->-
R2"
is
a
symplectic embedding of a starlike
a constant L > 0 satisfying
such that there exists
<p(z)\
are
a
We start with
and
\z
>
L\z-z'\
proceed along
U and cp
1. Reduction to
origin
:
for
all z,
z'
G
(C.2)
U.
has the extension property.
We shall first
sumptions
a
domain.
R2"
domain U C
on
X
do not know of
Lipschitz
pair (U, cp)
du
that there do exist starlike domains which
see
we
\cp(z)
Then the
curves
for all z, z'
domain U
<
Theorem C.6 Assume that cp
Lipschitz
of
inf {length (y)}
:=
domain U C R" is
schitz domains. But
length
0 such that
du(z, z)
Each
shall
we
|y'(,s)| ds.
distance function
dv(z, z')
<
Definition C.5 We say that
/
Jo
all smooth
over
of Theorem C.6
proof
the starlike domain U. The
by
:=
a
du(z,z')
and y (I)
on
Rn is defined
—>
in the
reasons
condition
cp in Theorem C.6 below
on
observing
that <p(0)
=
the lines of
sufficient to
simpler
that
we
0 and
the arguments
our as¬
through.
case
may
a
assume
that U is starlike with respect to the
Indeed, suppose that Theorem C.6
subset of U whose closure in R2" is contained
dcp(0)
holds in this situation, that A is
push
[6] and then verify that
—
id.
in U, and that U is starlike with respect to p
^
0
or
that
cp(p) ^
0
or
that
196
D
C. The Extension after Restriction
dcp(p) ^
:=
define the
id. For
w
R2"
e
symplectic embedding \Jr
f
Then
ijr(p)
D is
0 and
=
(D T-p)
(D T-p)
(D T-p)
o
(A) is
o
(U) and
y
:
[0, 1]
id.
=
-
set z
o
=
U with y
D~l(w)
(0)
from
u>
to
u/,
and
^
is starlike with respect to the
(U) whose closure in R2"
=
a
+ p,
z, y
(I)
DoT-poy: [0, 1]
runs
(U)
w.
We
R2" by
Since U is starlike with respect to p and
o
(U).
path
o
Assume next that U is
o
rw the translation zh-z +
(D r_p)
:
(D r_p) (U)
a subset of (D
t-p)
linear, the domain
by
t-tpip) ocpotpo D~l.
:=
dijr(p)
denote
we
Principle
A.-Lipschitz
z'
=
=
z',
-»
domain.
D~1(w')
and
is contained in
We fix
w,w'
G
+ p. Given any smooth
the smooth
(Do X-p)
origin,
path
(U)
so
<*(DoT_p)(t/) (">,«/)
f
<
\Dy'(s)\ds
\\D\\j
<
\y'(s)\ds.
It follows that
d(Dor„p)(U)(w>w')
<
\\D\\du(z,z')
<
\\D\\X\z-z'
.-l
IDHA.
Since
w,w'&
||D|| Il
D_11
(Do x-p)
À-Lipschitz
(U)
arbitrary,
were
domain.
\f(z)-f(z')\
>
Finally,
L
the
we
£>"
\w
—
w
conclude that
(D t-p)
assumption (C.2)
D~\z-z')
>
\D\
\z-
o
on
cp
(U)
is
a
yields
z
(Do r_p) ([/). By assumption we therefore find a symplecto¬
morphism V(DoT_p)(A) of R2" suchthat*(flor_p)(A)|(Dor_p)(A)
^|(dot_„)(A)for all
z,z'
g
=
Define the
symplectomorphism <J>A
<*>/i
Then
<&\a
=
cp\a,
as
:=
r^(p)
required.
o
of
R2" by
^(öor_p)(A)
o
D
o
t_p.
197
Step
2. The classical
approach
0 and
origin and that cp(0)
id. We denote the set of symplectic embeddings of U into R2" by
dcp(0)
Symp([/, R2"). Since U is starlike with respect to the origin we can define a
continuous path cpt c Symp (U, R2") by setting
So
that U is starlike with respect to the
assume
=
=
V,(Z)
\
:=
z
I !„(«)
path cpt is smooth except possibly
diffeomorphism n of [0, 1] by
The
the
e
=={«»,-«/.
a
smooth
path
in
Since U is starlike, it is
cpt(U),
t
:=
<6
0. In order to smoothen cpt,
t
we
0,
=
and
set
(C.5)
a
We have cpo
so
define
(C4)
[0, 1] and ze[/we
t G
Symp (U, R2n).
contractible,
(C3)
*>.!].
<Pm(z).
[0, 1]. We therefore find
G
if
0,
=
if.elO,.],
denotes the Euler number, and for
Then cpt is
t
if
<t>t(z)
sets
—
[0
»<»
where
at t
if
the
same
smooth
=
id\j
and cpi
=
cp.
holds true for all the open
time-dependent
Hamiltonian
function
(J
H:
{t}xcpt(U)
-*
(C.6)
R
te[0,l]
generating
the
path
<pt,
i.e., cpt is the solution of the Hamiltonian system
ftcpt(z)
<t>o(z)
Here,
=
JVHt(cpt(z)),
zeU,
=
z,
zeU.
J denotes the standard
complex
co0(z, w)
The function
H(z,t)
smooth function
h(t)
=
:
=
te
structure defined
(Jz, w),
[0,1],
(C.7)
by
z,wR2n.
Ht(z) is determined by the first equation in (C.7) up
[0, 1]
->
R. Notice that 0
G
cpt(U)
for all
t.
to a
We choose
h(t) suchthat
Ht(0)
=
0
for all
t e
[0, 1].
(C.8)
C. The Extension after Restriction
198
Step
3. Intermezzo: End of the
Before
Fix
a
proceeding
C.l
proof of Theorem C.6 we shall prove Proposition C. 1.
whose closure A in R2" is contained in U. Since U is
with the
subset A of U
bounded, the
proof of Proposition
Principle
set A is
compact, and
(J
K=
so
{t}
the set
cpt(Ä)
x
[0, 1]
C
x
R2"
te[0,l]
is also compact and hence bounded. We therefore find
V of K which is open in
[0, 1]
4>t(U). By Whitney's Theorem,
x
R2"
a
bounded
there exists
smooth function
a
and vanishes outside V. Since V is
to 1 on A'
Then
<Î>a is
with compact support, and
<$>a\a
resulting time-1-map.
<p\a- The
=
on
[0, 1]
x
x
R2"
Hamiltonian system
define Oa to be the
symplectomorphism of R2"
proof of Proposition C. 1 is thus
defined
globally
a
/
[Jt e[0 ^{t}
bounded, the function
equal
fH: [0, 1] x R2" -> R has compact support, and so the
associated with fH can be solved for all t g [0, 1]. We
which is
neighbourhood
and is contained in the set
complete.
Step
4. End of the
If the set U is not
compact, and
exists for all
C.6
bounded, the subset A
there
so
t
proof of Theorem
might
be
C
off
no cut
We shall first
that
carefully.
verify
is linearly bounded. Since we do
bounded gradient field to a linearly
our
relatively
of H whose Hamiltonian flow
fH
[0, 1]. We therefore need
G
U does not need to be
to extend the Hamiltonian H more
assumption (C.2)
on
cp
implies
that S7H
not know a direct way to extend a
bounded
gradient field,
we
linearly
shall then pass to
the function
H(t, w)
G(t, w)
where g
(\w\)
=
\w\ for \w\ large.
we
that G is
shall obtain
an
g(M)
Our
assumption
continuous in
Lipschitz
imply
function G
continuous
Lipschitz
will
=
extension H(t,w)
=
that U is
and
R2".
can
a
Lipschitz
x
After
bounded.
Lemma C.7 Let L
(i)
\cpt(z)
-
(ii) ||d<Mz)||
>
4>t(z')\
<
-
0 be the constant
>
L
far
\z
-
all
z'\
t G
domain
hence be extended to
a
smoothing G in w to G
g(\w\)G(t,w) whose gradient is linearly
[0, 1]
on
w
guaranteed by assumption (C.2).
for all
t G
]0, 1] and
]0, 1] and
z G
U.
z,
z'
G
U.
199
Proof, (i)
In view of definitions
(C.5) and (C.3)
(t>t(z)
for all
]0, 1] and
t
z G
\(t>t(z)
-
=
-^-<p(ri(t)z)
4>,(z')\
-\-\<p(r)(t)z)
=
-
T](t)
we
find
cp(n(t)z')\
I
ri(t)
,i
L\n(t)z-n(t)z'\
L\z-z'\.
=
(i) thus follows.
(ii) We fix
we
(C.9)
Together with assumption (C.2)
U.
1
[26]
have
»KO
~
Assertion
we
t
]0, 1] and
G
decompose
proof of Proposition
symplectomorphism dcpt (z) as
z
the linear
U.
g
Following
dcpt(z)
=
the
2.20 in
PQ
symplectic and P is symmetric and positive definite and
orthogonal. According to [26, Lemma 2.18] the eigenvalues of P are of the
where both P and
Q is
Q
are
form
o
Since
ai
<
a.2
<
Q is orthogonal,
<
we
...
a„
<
eigenvector
an
X\l
so
assertion
For
e
>
0
r
so
<
(ii)
>
<
...
=
||P ||
=
<
X-21
^T1-
-
X~l.
of Ai. In view of assertion
A-i \vi\
and
x"1
find
I|d0r(z)||
Let vi be
<
=
\dcpt(z)vi\
>
L
(i)
(CIO)
we
have
|t>i|
L~l. This and the identity (CIO) yield ||d0,(z)||
<
L~\
and
so
follows.
0
we
by B(r) the closed r-ball around
B(e) C U. Finally, we abbreviate
denote
small that
U,
=
Unß(-el/tY
ts]0, 1].
0
G
R2".
We choose
(Cll)
C. The Extension after Restriction
200
Lemma C.8
(i) There
exists
|VJÏ,(u;)|
(ii)
There exists
a
(i) Fix
definitions
t
—r-
tl
|w|
constant ci
\VH,(w)\
Proof,
Ci
<
]0, 1] and
(C.9) and (C.4)
JVHt(w)
we
all
for
w
G
cpt(Ut).
in
(C.7) and the
1
\-7^<P(ri(t)z)
\n(t)
/
n'(t)
1
/w/
=
=
]0, l]andw
cpt(z). Using the first line
=
(
-77
dt
t
t G
compute
d
(ii) with
]0,\]andw Gcpt(U).
t
-cpt(z)
dt
=
=
Lemma C.7
0 such that
>
for all
\w\
tl
Ci
0 such that
>
%e~llt
<
g
constant
a
Principle
r}(t) \
nit)
^(-w
+
cp(n(t)z)+dcp(n(t)z)z)
(C.12)
dcp(n(t)z)z).
(C.13)
yields
1
\\d(p(z)\\
<
j
(C.14)
for all zeU
Lj
and the
identity
cpt (0)
=
-^cp(0)
\w\
In view of the
identity (C.13)
\VHt(w)\
=
=
=
0 and Lemma C.7
\cpt(z)\
>
<
Ci
:=
2
(l
+
jj j
is
as
(C.14)
and
(C.15)
we
^(\w\ + \\dcp(n(t)z)\\\z\)
2
The constant
0
(
^(M
required.
yield
(C.15)
tL
<
=
L\z\.
and the estimates
\JVHt(w)\
(i) with z'
1
+
^M
conclude
201
(ii) By
lor's Theorem
constants
where
Since
applied
B(e)
to cp:
R2"
->•
Mi and M2 such that for each
and
<p(0)+d<p(0)x+r(x)
with
|r(x)|
dcp(x)
=
dcp(0)
with
\\R(x)\\
cp(0)
0 and
=
\cp(x)
and so, with
d<p(0)
dcp(x)x\
-
x
=
R(x)
the operator
=
\r(x)
we
Assume
dcp (ri(t)z)
(Mi
<
z
M2)n(t) \z\z
+
|x|2
,
M2\x\,
R(x)
dC(R2").
g
if
|x|
< e
if»7(0lzl<e-
Ut. In view of the definition (C.l 1) of Ut
»7(0 kl
A"2/,-«1/(
<
=
ee~xtte
<
we
(C.16)
then have
e.
e
Inserting
the estimate
(C.16) into (C.12) and using (C.15)
\VHt(w)\
<
<
Lemma C.9
:=
2(Mi
(i) There
\Ht(w)\
(ii) There
exists
a
\Ht(w)\
+
—
t2
(Mi +
<
a
tl
constant c2 >
<
C-2-e-l,t\w\2
tl
is
constant
-I'M2
M2)ee~l/te
1/f(Mi
as
+Af2)e—|iw|.
required.
C2
for all
\z
1
m,
-ê
M2)eej^
exists
conclude that
2
2
<
we
^(Mi+M2)r;(0|z|2
t
-
The constant ci
Tay¬
guarantees
r)(t)z,
-
now z G
<
\x\2
(Mi + M2)
<
1
(V(t)z)
so
conclude that
R(x)x\
-
Mi
<
of the linear operator
norm
id^n
=
dC(R2")
-»
=
+
B(e), and
B(e),
g
x
dcp: B(e)
on
<p(x)
||F(x)|| denotes
-
t G
0 such that
]0, 1] and
w
ecp,(U).
0 such that
for all
t G
]0, I] and
w
G
cpt(Ut).
202
C. The Extension after Restriction
Proof, (i)
Fix
/ e
]0, 1] and
y:
joins
0 with
w.
Since
w
[0, 1]
Ht (0)
Ht(w)
0
I
=
0, the
Using the identity (C. 17),
and
(C.15)
we can
cpt(sz)
I
(VHt(y(s)),y'(s))ds
Jo
(VHt(cpt(sz)),dcj)t(sz)z)ds.
value theorem and Lemma C.7
mean
\cpt(sz)\
=
find that
we
Jo
identity <pt (0)
y(s)
cpt(U),
Ht(0)+
=
=
The
<pt(z). The smooth path
=
-+
=
Principle
\cpt(sz)
=
Lemma C.8
-
(i),
cpt(0)\
<
Lemma C.7
(C.17)
(ii) yield
±-s\z\.
(C.18)
(ii) and the estimates (C. 18)
estimate
rt(w)\
f \VHt(cpt(sz))\\dcPt(sz)z\ds
<
Jo
Ci
<
t2
L
Ci
<
r1
i
-|z| / \<t>,{sz)\ds
Jo
1
2
1
t2
1
<
The constant C2
(ii) Assume
:=
5Clpp|«,|>.
\ Ci -h is
now
z
G
11,
as
required.
Ut. Using
Lemma C.8
(ii) and estimating
as
above
we
obtain
\Ht(w)\
The constant c2
Choose
a
:=
\ ci -h
is
as
<
Id-lê-^kl2.
required.
smooth function g:
[0,oo[—>• [l,oo[ such that
1
*'**'
r
if
g(r)
r
>
2
(C.19)
Figure
and 0
<
g'(r)
1 for all
<
62: The function
g(r).
\^jtero i]{t}
cpt(U)
r.
We define the smooth function G:
G(t, w)
Lemma C.IO
(i) There
|VG/(w)|
(ii)
There exists
a
<
<
forall
-=-
tl
:=
C3
t
>
(C
g(\w\)'
0 such that
&]0,\]andw &(pt(U).
0 such that
%e~xlt
for
all
t G
]0, \]andw
G
cpt(Ut).
We have that
g'(\w\)
1
g(\w\)J
and
Rby
H,(w)
Gt(w)
a constant
constant C3 >
|VGt(tü)|
Proof, (i)
exists
=
x
w
g(\M)2 Nl
so
VGt(w)
g'(\w\)
g(\w\)2
w
—-H,(w) +
\w\
1
gm)
VH,(w).
204
C. The Extension after Restriction
Using g'(t)
[0, 1], Lemma C9(i) and Lemma C8(i) and |iu|
g
Principle
g(|w|)
<
we
therefore find that
\VGt(w)\
—l—\Ht(w)\
<
g(\\)2
\w\|2j.+
<
g(\w\)2 t2
(ii)
mating
Assume
above
as
Ci
:=
now z
+
C2 is
as
Lemma Cll
\Gt(w)
(ii)
-
Proof,
y
a
:
Fix t
Lipschitz
[0, 1]
-+
is
exists
<
(ci+c2)^_1/'.
<
tL
required.
as
constant
a
-ttIw-w'I
C4
>
0 such that
for all
tl
t
e]0,\]andw,w'
G
cpt(U).
constant C4 > 0 such that
Gt(w')\
(i)
ci + c2
Gt(w')\
a
I
Ut. Using Lemma C.9 (ii) and Lemma C.8 (ii) and esti¬
g
(i) There
There exists
\Gt(w)
is
—
If
obtain
we
:=
Cll
g(\w\) t2
required.
|VG,(u;)|
The constant C3
l
(Ci+C2)i.
<
C3
g(\w\)
_1_C2I
<
-
The constant
-^—\VHt(w)\
+
<
G
ê~l/t \w
tl
]0, 1] and
domain with
U such that
w
w'\
=
Lipschitz
y(0)
length(y)
-
=
=
z, y
for all
cpt(z), w'
constant X.
(I)
=
=
]0, 1] andw, w'
cpt(z'),
and
We then find
G
assume
a
cpt(Ut).
that U
smooth
path
z' and such that
f \y'(s)\ds
Jo
t G
<
2X\z'-z\.
(C.21)
205
Using
C10(i), Lemma C7(ii),
Lemma
the estimate
\Gt(w')
Gt(w)\
-
i
=
l
(VGt(cpt(y(s))) ,dcpt(y(s))y'(s))ds
*
2z/W
<
-4-2X\z'-z\
<
'
'
t2 L
C3
1
-=-
—r
,
2k
The constant C4
path
Assume
:=
2C3J2X
now
z,z'
The constant cù,
Our next
f
a
a
Gt(w)\
-
on
[0, 1]
(Mc Shane
R
W
:
—>
X-Lipschitz
V U
-ê-1/t-^2X\w'
<
tl
x
Lz
(ii) and estimating
as
-
w\.
D
R2" having
[27]!)
Consider
R which is
:=
inf
on
Ure[0 i](0
similar
a
properties.
subset W
X-Lipschitz
{/(w)
continuous extension
function
Lipschitz
Lemma CIO
that the
required.
as
Lemma C.13 Assume that V is
that the
Using
we can assume
x
<Pt(U)
to a con¬
We shall need two
of the
metric space
continuous.
(X, d)
Then the function
defined by
J(x)
is
required.
lemmata.
function f
—>•
'
is to extend the function G
goal
Lemma C.12
X
w\.
Ut. Since £/ is starshaped,
G
2c?, -kx is
:—
tinuous function G
auxiliary
-
'
obtain
we
\Gt(w')
:
as
y chosen above is contained in Ut.
above
and
is
,
,
\w
t2 L2
(ii)
(C.21) and Lemma C7(i)
estimate
we can
h
V U
:
continuous
on
B(2r)
B(2r).
Xd(x, w) \
+
w
Urs
Lang
for
a
subset
—>
R is
of R2"
which contains the
Xy-Lipschitz continuous
(2Xy + Xß)-Lipschitz
Then h is
pointing
W}
of f.
B(r).
lVm grateful to
g
out to me this reference.
on
origin and
V and Xß-
continuous
on
206
C. The Extension after Restriction
Proof. Fix w,w'
VU
G
B(r). If
\h(w)
So
assume
that
V
e
w
h(w')\
—
e
V and 0
G
i
<
B(2r)
so
h is
(2Xy
+
a
i
\w
<
B(2r) then by assumption
e
\w
\w'\
w
—
—
w'\.
< r
<
^
and
so
/i
\
.
estimate
\h(w')-h(0)\
\h(w)-h(0)\
<
Xy \w\
<
(2Xy+XB)-Y
(2Ày + Xß) \w
w'\
+
\w'\
+ Xß
\w\
—
ÀB)-Lipschitz
Lemma C.14 There exists
w'
<
<
and
—
we can now
\h(w)-h(w')\
or w,
B(r). Then
G
i
/i
\w \
i
\w\
V
G
max(A.y, Aß)
<
\ B(2r) and w'
M
Since 0
w'
w,
Principle
continuous
on
V U
continuous function G
,
B(r).
[0, 1]
:
D
x
R2n
—>
R with the
following properties.
(i) G(t, w)
(ii) There
=
exists
\Gt(w)
constant
Gt(w')\
<
C5
>
-4 \w
(i) and (ii), and shall then
a
[0, 1] and
t G
w
G
(pt(U).
0 such that
w'\
-
tl
We shall first construct
Proof.
tions
-
a
all
for
G(t, w)
for all
function G
verify
[0, 1]
:
t G
x
]0, 1] andw, w'
R2n
->
R
G
meeting
R2".
asser¬
that G is continuous.
We set G(0, w)
0 for all w G R2". Since H : \Jt
x cpt (U) ^ R is
6 [0^j^}
continuous, Lemma C.9 (ii) and the definition (C.l 1) of Ut imply that H(0, w)
=
=
0 for all
all
w;
g
We
w
G
U. In view of definition
U, and
now
fix
so
assertion
t g
(C20)
(i) holds for
t
=
we
therefore have
G(0, w)
=
0 for
0.
]0, 1]. We define the number Rt by
Le
Rt
=
1
/f
--el't.
2
e
(C.22)
207
Fix
w
=
cpt(z)
B(2Rt).
G
In view of the estimate
(C.15)
and the definition
(C.22)
have
we
Izl
and
<
2
^ TRt
\w\
V
L
e
i,,
~e
=
L
>
e
Ut in view of definition (C. 11). Lemma Cll (ii) therefore implies that
Gt is ^fe_1^-Lipschitz continuous on cpt(U) D B(2Rt). According
so z
the function
to Lemma C12
Gt(x)
is
a
:=
the function Gt
inf
\Gt(w)
jfe_1/r-Lipschitz
B(2Rt)
:
ye~1/f
+
extension of
\x
-
Gt
R defined
->-
\w
w\
to
e
by
cpt(U)
B(2Rt).
In
D
B(2Rt)\
(C.23)
particular, the
function
Gt: cpt(U)UB(2Rt)^R,
5,C*):={S?> *"*<£
if
[ Gt(x)
is well-defined.
According
<pt(U), and according
on
According
to
to Lemma
to the
we
C5
Lemma C.12
Gt(x)
is
a
:=
inf
once
{ Gt(w)
-f-Lipschitz extension
The function G
:
]0, 1]
x
more,
+
%-Lipschitz
jf-Lipschitz
continuous
<pt(U)
=
R2"
on
B(2Rt).
abbreviated
2C4 +
:=
C4.
find that the function
we
C5
-y |x
-
G(t, w)
—>•
continuous
B(Rt) is therefore
U
w\
|
w
e
cpt(U)
U
for all
w
G
Gt
R2n
:
\
B(Rt)
of the restriction of Gt to cpt (U) U
G(t, w)
t G
C.ll(i), Gt is
above, Gt is
(C.24)
B(2Rt),
Lemma C.13, the restriction of Gt to
-4 -Lipschitz continuous where
Applying
fined by
G
x
—>
R de¬
(C.25)
B(Rt). In particular,
cpt(U).
R thus defined therefore meets assertion (i) for
]0, 1] and assertion (ii).
We
in the
are
left with
showing
that the function G
:
[0, 1]
x
R2"
-*
R constructed
previous two steps is continuous. The definitions (C.22), (C.23), (C.24) and
(C.25) show that the functions G( •, x) : ]0, 1] - R and G(t, ) : R2n - R are
208
C. The Extension after Restriction
Figure
63: The domain
cpt(U)
and its intersections with
continuous. This and the fact that the functions
imply
that G
:
]0, 1]
continuous at
t
R2"
are
B(Rt) and B(2Rt).
%-Lipschitz continuous
R is continuous. In order to show that G is also
-^
w) for each
(0,
R2" centered
ball
/
x
G(t, )
Principle
w
e
R2"
we
fix
w
e
R2n.
We choose
an
open
at w. In view of definition (C.22) we have Rt —> oo as
Bw C
-* 0+. We therefore find 10 > 0 such that Bw c B(Rt) for all t G
JO, t0]. We fix
G ]0, to] and w' e Bw. Recalling the definition of G(t, w')
Gt(w') we see
=
that
Gt(w')
Since 0
=
=
^t(w')
cpt(0)
G
=
Gt(w')
=
inf
cp,(U)
n
\Gt(v) + C-\e~xlt \w'
B(2Rt) and Gr(0)
G,(«/)
Moreover,
we
recall from the
c4e_1/f-Lipschitz
continuous
|G,(u)|
<
CA
<
C4re~x/t
cpt(U)
\v\
v\ \
Ht(0)
g
v
=
0
cpt(U)
we
n
n
_1
of the
proof
B(2Rt).
for all
v G
B(2R,)\.
conclude that
(C.26)
^«-Û'l.
beginning
on
=
-
of Lemma C.14 that
This and
Gt(0)
0,(^)
B(2Äf).
H
=
0
Gt is
yield
209
Therefore,
Gt(v)
+
4e~l/t\w'-v\
-\Gt(v)\
>
tL
4re-x/t lu/- ul
+
tl
C-±e-xlt(-\v\
>
+
t2
ca
>
-i/t
i
'
—r-e
'
t2
for all
&(£/)
G
v
PI
ß(2#,). We
imply
>
-C4:e~x/t \w'\
t2
(C.26) and (C.27), which hold for all
'
(C.27)
.
]0, to] and u/
t g
g
#„,,
now
that
\Gt(w')\
and
/i
conclude that
Gt(w')
The estimates
u>
\w' -v\)
<
CAre~xlt \w'\
G is continuous at
so
Let
also the
A be
now
origin
a
for all
t G
]0, f0] and u/
G
5«,
(0, w). This completes the proof of Lemma C. 14.
subset of U whose closure in
is contained in »J,
we can assume
R2" is contained
in U.
that A is closed and 0
G
Since
A. We
abbreviate
A
U
:=
{t]xcpt(A).
te[0,l]
The next
step is
to
smoothen G in the variable
function G coincides with G
which
approximates
on
w
in such
way that the smoothened
a
A. We shall first construct
G very well and shall then obtain G
a
smooth function G*
by interpolating
between
G and G*.
Since
R2"
is
a
normal space,
we
find
an
open set V in
R2" such
that A C V C
V C U. Then
0KA)
C
0,(V)
C
cpt(V)
=
0,(V)
C
</>,(£/)
for all ?
G
[0, 1].
(C.28)
We abbreviate
V
\J
:=
{t}
x
cpt(V).
'[0,1]
Since A is closed and V is open in
/
:
[0, 1]
x
R2"
-+
[0, 1]
x
R2",
we
find
a
smooth function
[0, 1] such that
/U
=
l
and
/|[0,i]xR2«\v
=
0.
(C.29)
210
C. The Extension after Restriction
We say that
variable
w
continuous
continuous function F:
a
R2"
G
on
[0, 1]
x
is smooth in the variable
(ii)
\G*(w)
\VG*(w)\
Proof. For
-
tl
each /
Vi
Then there exists
N
g
a
:=
continuous function G*
a
R2"
w G
-4
<
R2"
x
[0, 1]
:
Gt(w)\
<
for all
t G
-4
far all
x
R2",
{(f, w)
]0, 1] and
[0, 1]
G
partition
0,-
w G
x
:=
{j
Mi
Since the functions
Pi
{supp 0,}
N | suppö,- n
G
w
G
R2".
on
form
x
<
/}
[0, 1]
a
R2" by
1]
R2"
x
(C.30)
.
such that for
V/. We let /,• be
some
locally
finite
covering
a
of
We next choose
a
suppfy yé
0}
m(-. We set
:=
{my
max
:=
max
{|V0/(u;)|
positive
smooth
K(v)dv
=
/c
|
\ j
(t, w)
numbers r,
n
/R2n
R which
e
®{}.
(C.31)
$,• have compact support, the numbers
finite. We define
5(1) and
-»
R2".
R2" | \Vft(w)\
unity {6i)ie^
of
Since
V/,.
c
finite; let its cardinality be
are
are
the set
©,is
exist and
R2"
x
]0, 1] and
t G
each i the support supp Qi is compact and contained in
[0, 1]
w
and has the following properties.
define the open subset Vi of [0,
we
smooth
number such that supp
R is smooth in the
->
of F with respect to
R2".
Lemma C.15 There exists
(i) |V/,(iü)|
[0, 1]
DkFt(w)
if all derivatives
Principle
bump
:=
G
[0, 1]
x
—|—.
(C.32)
(C.33)
h Mi pi
function K
max
+ 1
by
:
R2"
—>
[0, oo[ such that supp K
1. We abbreviate
:=
R2"}
j|VÄ"(u)| | ugR2"}
c
211
For each i
we
define
a
smooth function Ki
:
R2"
—>
[0, oo[ by
K^--=jnK{j-y
Then supp Ki C
f^n
jB(r;) and
|V^(u;)|
<
Ki(v)dv
1, and
=
^-r/c
r2n+l
for all
u; g
Let G be the function
function G*
:
[0, 1]
G*(t,w)
Since for each
to \->
t
G*(£, u>)
guaranteed by Lemma C.14.
R2" -> R as the convolution
x
(Gt*Ki)(w)
:=
=
f
Jr2"
R2".
(C.34)
For each i
define the
we
Gt(v)Ki(w-v)dv.
(C.35)
the function Gt is continuous and since Ki is smooth, the function
is smooth and
DkG*(t,w)
f
=
Gt(v)DkKi(w-v)dv,
k
=
0,l,2,...
(C.36)
JE.2"
(see, e.g., [16, Chapter 2, Theorem 2.3]). The function G is continuous, and
is continuous and has compact support and is thus
(C.36)
therefore shows that
continuous and smooth in
defined
DkG*
is
uniformly continuous.
continuous, k
—
0, 1, 2,..., and
It follows that the function G*
w.
[0, 1]
:
x
DkK{
Formula
so
R2"
G* is
->
R
by
G*(t, w)
J2ei(f' w)G*(t, w)
:=
(C.37)
i
is continuous and smooth in
t g
id.
In order to prove assertions
(i) and (ii)
we
fix
]0, 1] and abbreviate
6i(w)
ei(t,w),
=
G(w)
Proof of (i).
Using
in2« Kj
=
(v) dv
1
=
G(t,w),
the definition
we
G*(w)
=
G*(t,w),
G*(w)
(C.35) of the function G*-
=
=
f
(G(v)-G(w))Kj(w-v)dv
f
(G(w
-v)-
G*(t, w).
and the
find
G*(w)-G(w)
=
G(w)) Kj(v) dv
identity
212
C. The Extension after Restriction
and so,
together
with Lemma C. 14
G*(w)-G(w)
Principle
(ii),
f
<
|G(w
G(w)|
Ki(v) dv
^-\v\Kj(v)dv
/
<
-v)-
JB(t,) t2
C5
<r
j
I
Kj(v)dv
J B(n)
9
t
If
V/f (iu)
0, assertion (i) is obvious. So
=
the definitions
(C.33)
(C.31)
and
(C.32) that Mj
of Yj, the inclusion supp
1
rJ
(C.38)
2rJ-
C
1
<
=
IjMjpj
The definition
9j
~
V/;
assume
1 and pj
>
1.
>
and the definition
>
0. Recall from
This, the definition
(C.30)
of V;
yield
1
for all
<
—
Ij
~
\G*(w)-G(w)\
w
g
\Vft(w)\
of G* and the estimates
(C.37)
|V/f(iy)|
=
(C.38)
and
supp
(C.39)
6j.
now
(C.39)
yield
^07(w)(g*(u;)-gW)
j
<
J2ejw
c5
1
c5
t2 \Vft(w)\
1
t2 \Wft(w)\
and
so
assertion
(i) follows.
PYoof of (ii).
Using
J] 9j(w')
1
•
=
we
the definition
(C.37) of G* and the identities ^ 9j(w)
•
compute that for all w,w'e
G*(w')-G*(w)
=
R2n,
^0y(u/)G*(u/)-;£0;(u,)G*(u;)
j
=
=
i
£ (eJ(u/) -0j(w))(G)(w')-G(w'))
j
+
J2 9j (w)
(G* (u;') -G*j{w)).
(CAO)
213
Fix
now
Bw
c
mean
We choose i such that
w.
R2"
centered at
w
such that
and the estimate
(C.38)
<
Mj
J
in view of the definition of m,.
1 and
open ball
find that
we
Pj\w'-w\
<
(C.41)
9
*
>
(C.42)
l2rJt
whenever j
mi
®i, and
g
so
je®,
The definition
(C33)
of
Yj and the inequalities
(C43) yield
Ew
E^p^-< E^L
=
je©,
Since w, u/
estimates
G(w')
-
(C.31) of Mj implies that Mj
je®,
/;-
Bw.
e
an
In view of the
replaced by w' yields
w
G*(w')
The definition
c supp 0(-.
(C.32) of pj
choose
we
Fix w'
|V0,-(u)| |u/-u;|
max
veBw
with
0, and
>
Bw
value theorem and the definition
\9j(w')-9j(w)\
>
9i(w)
G
Bw
c
supp0(-
(C.41), (C.42)
J2 (OjirV)
-
and
9j(w))
have
we
"*•»
J
je®,
(C.44)
9j(w')
now
—
9j(w)
=
(C.44)
J
je®,
0 if
j g ©,-. This and the
show that
(g*(w') G(w'f)
<
-
C5
J2pj\w'
w\
-z-rj
je®,
<
\w
-z-
~
—
w\
'
r
Next, the definition (C.35) of G*j and the identity
G*(w')-G*(w)
/
with Lemma C. 14 (ii)
G*(w')-G*(w)
J
]
<
=
1
yield
f G(v)(Kj(w' -v)-Kj(w-v))dv
=
=
Together
f^n Kj(v)dv
(C.45)
.
(G(w'
we
-4
t2
-v)- G(w
-
v)) Kj(v) dv.
obtain
[
yR2„
\w'-w\Kj(v)dv
'
=
-?-\w'-w\
t2
214
and
C. The Extension after Restriction
so
YjOj(xv){g*(w')-G)(w))
The
identity (C.40)
and the estimates
(w )
Since w'
Bw
G
arbitrary,
was
G
—
we
(C.45)
(w)\
so
assertion
smooth in the variable
w
G
a
R2"
Let
/: [0, 1]
C.15,
4>t(U)
we
VGt(w)
w
verify
=
imply
w\.
2C5
<
x
R2"
t G
>
:
[0, 1]
x
R2"
:=
for all
imply
w
e
Gt
=
t g
+
(1
that G is smooth in
Vf(w)
Gt, and
(Gt(w)
R which is
cpt(A).
]0, 1] and
and Lemma C.15. We define
(ii)
—>•
w
e
we
R
2n
.
[0, 1] be the smooth function chosen before
f(t, w)G(t, w)
assertion
D
0 such that
-f-
-+
[0, 1] and
-
(C.28) and the identities (C.29),
G* is smooth in
have
—
now
(C.46)
w\
and let G and G* be the continuous functions
G(t, w)
In order to
—r-\w
—
and has the following properties.
Cc
<
guaranteed by Lemma C.14
G: [0, 1] xR2" - Rby
The inclusions
C(,
constant
VG,(u>)
Lemma
(C.46)
continuous function G
a
for all
G(t, w)
=
(ii) There exists
Proof.
and
—z-\w
(ii) follows. The proof of Lemma C. 15 is complete.
Lemma C.16 There exists
(i) G(t, w)
<
<
conclude that
VG*(w)
and
Principle
fix
t G
w
a
[0, 1]
on
R2"
x
continuous function
f(t, w))G*(t, w).
Lemma
C14(i) and the
and that assertion
]0, 1]. We first
fact that
(i) holds
assume w G
true.
cpt(U).
On
so
-
G*(w)) + ft(w)VGt(w) + (l
-
ft(w))VG*(w).
215
In view of Lemma C15
(i), Lemma CIO (i) and Lemma C. 15 (ii)
we can
therefore
estimate
|VG,(uO|
|V/,(u;)||G?(u;)-GKu;)|
<
C5
<
—
t2
We next
assume w g
C3
2C5
t2
t2
-]—± -1
We
:=
Cj,
+
|VGKu;)|
|VG*(u;)|
+
_
R2" \cpt(V).
\VGt(w)\
Setting C(,
plete.
+
R2" \cpt(V)
On
\VG*(w)\
=
have
we
ft
=
0, and
so
2C5
<
t2
'
3Cs assertion (ii) follows. The proof of Lemma C.16 is
com¬
D
in
define the desired extension H of H. Let g be the
function chosen in (C.l9) and let G be the function guaranteed by Lemma C.16.
are now
a
position to
Lemma C.17 The continuous function H
H(t, w)
is smooth in the variable
(i) H(t, w)
a constant
\VHt(w) \<
Ht(w)
:=
R2" and has
C
all
-
t e
the following
[0, 1] and
forall
l)
In order to
(C.47)
we
assertion
(ii)
R
defined by
(C.47)
properties,
(pt(A).
w e
e]0, \]andw
t
continuous, and since g(r)
of H, from Lemma C.16
verify
—>
[1, oo[ is smooth and G: [0, 1]
the function H is indeed
(C.47)
R2"
g(\w\)Gt(w).
G is smooth in w, the function H is smooth in
definition
x
0 such that
>
-(\w\ +
tl
Since g: [0, 00[
continuous,
G
for
H(t, w)
=
(ii) There exists
Proof
w
=
[0, 1]
:
we
fix
t
G
w.
Assertion
(i) and from
]0, 1] and
(i)
G
x
=
R2".
R2"
-
1 if
<
G
r
R is
\
and
follows from the
the definition
w
(C.48)
(C.20)
of G.
R2". Using definition
compute
VHt(w)
=
g'(\w\)^-Gt(w)
\w\
+
g(\w\)VGt(w).
(C.49)
216
C. The Extension after Restriction
Since 0
A and
G
cpt(0)
—
0
have, together with equation (C.8),
we
Gt(0)
This, the
mean
Using
g(r)
the
<
-l\w\
\VHt(w)\
C
=
Ht(0)
estimates
<
'
>
0
(C.50)
we can
=
0.
(ii) yield
|VG,(u;)|
and
tl
identity (C.49), the
holding for all r
'
Setting
<
+ 2
r
Gt(0)
=
value theorem and Lemma C16
\Gt(w)\
Principle
<
~4-
(C.50)
tl
and the estimates
|g'(r)|
<
1 and
estimate
9lwl + (lwl+2)9
t^
^(N + 1).
=
t^
tL
2Ce assertion (ii) follows. The proof of Lemma C. 17 is complete,
:=
Theorem C.6 is
field VHt(w)
on
locally Lipschitz
a
consequence of Lemma C.17: The
[0, 1]
x
R2"
continuous in
vector
is continuous, and since it is smooth in w, it is
w.
This and assertion
that the Hamiltonian system associated with H
We define O^ to be the
time-dependent
ü
resulting time-1-map.
can
(ii) of Lemma
C.17
be solved for all t
Since
VHt(w)
G
imply
[0, 1].
is continuous and
smooth in w, the map Oa is smooth
globally
that Oa
defined
\a
=
(see [1, Proposition 9.4]), and so <&a is a
R2". Moreover, Lemma C.17 (i) shows
symplectomorphism
of
<p\a- The proof of Theorem C.6 is finally complete.
Remark C.18
Proceeding
as
in
HA
which generates the
ever,
Ha might
not
Step
:
2
[0, 1]
we
x
obtain
R2"
-
a
smooth Hamiltonian
R
H U. How¬
symplectomorphism <E>a and is such that Ha U
be C°-close to H, and V/7a might not be linearly bounded.
=
D
Computer
programs
All the Mathematica programs of this
appendix
be found under
can
ftp://ftp.math.ethz.ch/pub/papers/schlenk/folding.rn
For
convenience, in the programs (but
capacity-axis
D.l
We fix
rescaled
are
by
the factor
not in the
text) both the w-axis and the
Xjit.
The estimate sEB
and «i and try to embed
E(it,a) into B4(2ti
2n/a)ui) by
multiple folding procedure specified in 3.3.1. If ui is admissible, we set
27T + (1
a otherwise.
A(a, ai)
2n/a)ui and A(a, «i)
a
+
(1
—
the
=
A[a_,
=
—
ul_]
:
=
Block[{A=2+(l-2/a)ul},
i
=
ui
=
ri
=
a-ul-ui;
li
=
ri/a;
2;
(a+1)/(a-l)ul-a/(a-1);
While[True,
Which[EvenQ[i],
If[ri
<
ui,
Return[A],
If[ui
<=
21i,
Return[a],
i++;
ui
=
a/(a-2)(ui-21i);
ri
=
ri-ui;
lj
=
li;
li
=
ri/a
]
]
,
OddQ[i],
If[ri
<
ui+1j,
218
D.
Computer
programs
Return[A],
i + +;
ui
=
ri
=
ri-ui;
li
=
ri/a
(a+1)/(a-1)ui;
3
]
]
]
This program
does what
just
whether ui is admissible
or not.
do not check whether the stairs
we
proposed
to do
in 3.3.1 in order to decide
Note, however, that in the Oddq [ i ] -part
Si+i
TA(A).
contained in
we
This
negligence
recognized to be
non-admissible in the subsequent EvenQ [ i+1 ] -part. Indeed, recall that 5,-+i Çl
T4(A) is equivalent to k+i > m(+i; hence r,-+i
(a/jt)li+i > «,-+i and Ui+i <
2/(+idoes not
cause
are
troubles, since if 5,-+i <£.
T4(A),
then «i will be
=
We denote the infimum of the admissible
A(a, ui)
increasing
we
have
accuracy
uO
for «i
a
=
uo, and A(a, ui)
<
function for «i
A(a, uq)
>
uq.
A(a, a/2)
<
«i's again by
=
2n +
Since, by (3.8),
=
rt
+
a/2
<
(1
—
uq
=
2n/a)ui
we can assume
Therefore,
a.
uq(o). Then
is
that uq
«o is
a
linear
<
a/2,
found up to
acc/2 by the following bisectional algorithm.
[a__,
acc_]
:
=
Blockt{},
b
=
c
=
ul
a/(a+1);
a/2;
=
(b+c)12;
While[(c-b)/2
If[A[a,ul]
ul
a,
c=ul,
b=ul];
(b+c)/2
=
]
acc/2,
>
<
;
Return[ul]
]
Here the choice b
the
resulting
sEB[a_,
estimate
acc_]
=
ait
/(a + tc) is also based
seb(ü)
:=
2
+
is
on
(3.8). Up
given by
(1-2/a)uO[a,ace].
to accuracy ace,
D.2. The estimate sec
219
D.2
The estimate sEc
We fix
a
>
yield
an
Ul
\a+îî' I [•
by
a
As
it.
have
optimal embedding.
seen
ul_]
following
in Remark
3.15.1, folding only
once
does not
We therefore fold at least twice and hence choose
ê first calculate
and «i. The
h[a_,
we
me
height
of the
image of T(a, it) determined
by looking at Figure 25.
program is best understood
:=
Block[{ll=l-ul/a},
i
=
ui
=
ri
=
a-ul-ui;
li
=
ri/a;
hi
=
211;
2;
(a+1)/(a-l)ul-a/(a-1);
While[ri
>
ul+11
-
li,
i + +;
ui
=
ri
=
ri-ui;
lj
=
li;
li
=
ri/a;
(a+1)/(a-l)ui;
If[EvenQ[i],
]
hi
=
hi+21j]
;
Which[EvenQ[i],
hi
-
hi+li,
OddQ[i],
hi
=
hi+Max[lj,21i]
];
Return[hi]
]
explained in 3.4.1, the optimal folding point uo(a) is the M-coordinate of
unique intersection point of the graphs of h(a,ui) and w(a, ui). It can thus
found by a bisectional algorithm.
As
the
be
uO[a_,
acc_]
:=
Blockt{},
b
=
c
=
ul
a/(a+1);
a
=
/2
;
(b+c)12;
220
D.
While[(c-b)/2
If[h[a,ul]
ul
=
>
>
Computer
acc/2,
l+(l-l/a)ul,
b=ul,
c=ul];
(b+c)/2
];
Return[ul]
]
Up
to
sEC[a_,
accuracy ace, the
acc_]
:=
resulting
estimate
sec(®) is given by
1+(1-1/a)uO[a,ace].
programs
References
[1]
H. Amann.
analysis,
Berlin,
de
Ordinary differential equations..
Gruyter
An introduction to nonlinear
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Lebenslauf
Ich wurde
Uitikon
15.
September 1970 als Sohn von Elisabeth und Franz Schlenk in
(Kanton Zürich) geboren. Die Primarschule besuchte ich von 1976-1982
am
in Uitikon. In Zürich besuchte ich anschliessend die Kantonsschule
der ich im
Wiedikon,
an
September 1989 die Matura Typus A bestand.
begann ich das Studium der Mathematik an der Eidgenössischen
Im Herbst 1990
Technischen Hochschule
(ETH) in Zürich, welches ich 1995 mit dem Diplom ab-
schloss. Prof. Bruno Colbois leitete meine
Diplomarbeit
über ein Thema
aus
der
Spektral- und der Kontaktgeometrie. Nach einem Besuch an der Université de
Chambéry kehrte ich anfang 1996 an die ETH zurück. Dort begann ich mit der
vorliegenden Doktorarbeit unter der Betreuung von Prof. Eduard Zehnder.

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