HOUSTON JOURNAL OF MATHEMATICS
@ 1998 University of Houston
Volume
THE
ISOENERGY
INEQUALITY
FOR
JAIGYOUNG
CHOE
Communicated
Let u be a harmonic
ABSTRACT.
nonpositively
alas.
by Robert
curved manifold,
no curvature assumption
4, 1998
A HARMONIC
MAP
M. Hardt
map from a unit ball B in Rn
into a
E(u) the energy of u, E(u\a~) the energy of
Then we obtain a relationship
isoenergy inequality, (n - l)E(u)
24, No.
between E(u) and E(ula~),
called the
5 E(U\aB). When the target manifold has
and u is stationary, it is shown that (n - 2)E(u)
5
are sharp because equality is attained
E(ulae).
These isoenergy inequalities
by some canonical harmonic maps.
1. INTR~DuOTI~N
It is the isoperimetric
with the volume
n>2
inequality
of the boundary
that relates
of D. Steiner
nnu,VoZume(D)n-l
where w, is the volume
D is a ball.
inequality
Recently
D in R”
VoZume(8D)‘“,
ball in R” and equality
holds if and only if
[7] and Croke [2] obtained the same isoperimetric
in a nonpositively
curved n-dimensional
simply connected
manifold
for n = 3 and 4, respectively.
curved Riemannian
manifold,
for dimension two (see [I$]).
In this paper we study
Let us consider
of a domain
Kleiner
for domains
Riemannian
of a unit
5
the volume
[13] proved that for any dimension
a smooth
Rm,n
2 2. Define
energy
of the restriction
E(u)
the sharp
an isoperimetric
harmonic
and Eat)
isoperimetric
inequality
But
in a nonnegatively
inequality
for energy instead
map u from a closed unit
to be the energy
of u to dB, respectively.
is known
Then
only
of volume.
ball B c R” to
of the map 2~ and the
is there any relationship
between E(u) and E(ula~)
that resembles the isoperimetric
inequality?
Here we
answer this question affirmatively:
we obtain a relationship
in a sharp form, called
Supported
in part by BSRI-94-1416,GARC,KOSEF(961-0104-020-2).
649
JAIGYOUNG
650
the isoenergy
inequality,
N is nonpositively
for a general
curved,
target
holds when N = R”
any Riemannian
manifold
<
if
E(u18B),
and u is a linear
of dimension
map.
Second,
when N is
> 3, we prove
(n - 2)E(u)
5
E(ulaB),
holds if N = ,YP1 c R” and u(z) = z/IzI.
where equality
We would like to thank
R. Schoen and L. Simon for their interest
2. THE ISOENERGY
Assume
N as well as Rm. First,
manifold
then we show
(n - l)E(u)
where equality
CHOE
that
M”,
in this work.
INEQUALITY
N” are Riemannian
manifolds
with Nk isometrically
em-
bedded in R”. We look at a bounded map u : M + N whose first derivatives
are in L2; such a map is thought of as a map u = (‘~11,. ., u,) : M t R” having
image almost everywhere in N. Then the energy E(u) of u is defined by
E(u)
I
=
M
where
IVu12 = C~=“=,IVui12, Vui being
lW2,
the gradient
of ui on M.
IVu12 is called
the energy density of u. The critical points of E(u) on the space of maps
referred to as harmonic maps. Thus u E C2 is harmonic if and only if
AM
(I)
u
I
are
T,N.
A harmonic map u is stationary
if its energy is critical with respect to variations
of the type u o Ft, where Ft : M + M is a smooth path of diffeomorphisms
of M
fixing the boundary.
It can be shown that a Co harmonic
that stationary
harmonic
maps satisfy the monotonicity
invariant
energy
following
lemma.
Lemma
2.1.
in balls.
We state an equivalent
(n - 2) L
The
harmonic
IVu12 =
monotonicity
2-n
P
form of the monotonicity
in the
Let B, = {x E R” : 1x1 < p} and B = B1. Suppose u : BI+~ +
N, E > 0, is a stationary
PROOF.
map is stationary
and
property for the scale
[Vu12
s
BP
formula
-
a2--n
map.
S,,
We have
( IVu12 - 2 1$4’)
[9,11] says
, T = 121.
THE ISOENERGY
for Cl< 0 < p < I+E.
the formula
FOR A HARMONIC
Noting that SaBP f = & s,
with respect
The first isoenergy
map of Bl+,
INEQUALITY
P
to p and set p = 1.
inequality
into an arbitrary
of this paper
target
MAP
651
f for almost all p, differentiate
cl
holds for a stationary
harmonic
manifold.
Theorem 2.2. Let n 2 3 and suppose that u : B1+E + N, E > 0, is a stationary
harmonic
Then
map.
(n - 2)E(ub)
I
E(uIas),
where equality can be attained if N = 57-l
Let vu,
PROOF.
denote
the gradient
C
Rn and u(x) = x/(x\.
of pi on dB.
Observe
that
It follows from (2) that
(3)
which gives the desired
inequality.
If u(z)
holds.
Remark
1. i) It should be mentioned
hold for nonstationary
harmonic
ii) We should remark,
in relation
smooth
harmonic
that
maps.
n = 3,4,5,6,
2.2, J.C.Wood’s
{&}
= (x,0)
inequality
curved space (Theorem
case is different
(n - l)E(u)
theorem
that any
is constant
[14] (see
of C2 harmonic
for z E dB, E(&)
for a harmonic
inequality
maps 4% : B +
< E(&+I),
and
map from B into R”.
for harmonic
2.4), we state it independently
and interesting
Theorem 2.3. Suppose that u is a smooth harmonic
Then we have the isoenergy
on 3B
iqf ‘LTk;).
2
it is a special case of the isoenergy
proof of the Euclidean
2.2 fail to
maps is still open.
there is a sequence
Now we prove the isoenergy
into R”.
= 0 and hence
See [8,10] for such maps.
n - 2 =
nonpositively
l%/drl
2.1 and Theorem
map u (n > 2) which is constant
S” c Rn+l (see [12]) such that &(x)
Although
Lemma
to Theorem
also [15]); the case for weakly harmonic
iii) When
= x/1x1, then
0
equality
inequality
F
E(ula~),
maps into a
because
the
in its own right.
map from fi C R”, n > 2,
JAIGYOUNG
652
CHOE
where equality holds if and only if u is a linear
PROOF.
map from R” to R”.
(1) implies
Aui = 0, i = 1,. . ‘,m.
Hence
Here, without
loss of generality,
let us assume
2Li =
0, i = l,...,m.
s aB
Using (3) and the fact that
one sees that
(*) 5
n - 1 is the first eigenvalue
the inequalities
EW2
I
above one gets
[E(ulaB)
-&E(ula~)
which gives the desired isoenergy inequality.
if ui is a constant multiple of dui/dr
and
f&B ui + (n - l)Ui
which holds if and only if u is a linear
=
curvature,
Moreover
equality
formula
I
[3] says that
~AlVu12 = lIV’dul12
0, i = 1,. . ., m,
map from B
c
0
R”, n 2 2, to N of
-
E(+B).
if u : M” + N” is harmonic
+
then
CRN(u,e,,u*ea,u,e~,u,ea)
a,B
(4
holds if and only
then
(n - l)E(u)
The Bochner
2)E(u)l,
- (n -
map from R” to R”.
Theorem 2.4. If u is a smooth harmonic
PROOF.
on 8B,
[~~~~~~u,12]1’2:L.(I18zj-(n-2)E(u)~1’2.
Hence by combining
nonpositive
of the Laplacian
C
RiCM(U*C9i,U*O,)
THE
ISOENERGY
where V’ is the pullback
INEQUALITY
curved,
for T*N.
]Vu12 is subharmonic.
653
basis for
Hence for M = B and N nonpositively
Since the mean value of a subharmonic
on a sphere of radius r centered
a function
MAP
from TN, el, ... , e, is an orthonormal
connection
TM and Or, ‘.‘, 01~is orthonormal
FOR A HARMONIC
at the origin is monotonically
function
nondecreasing
as
of r, one can deduce that
_
W7l
where equality
nwn
s
aB
follows from (2).
So
2
(5)
E(u)
/I I
<
i3B
Then
adding
Remark
(3) to (5) gives the isoenergy
2.
In case u is a harmonic
nonpositive
When
theorem
curvature,
the target
manifold
harmonic
map u E C2(B,
This theorem
can obtain
inequality.
of radius
p into N of
have
N is nonpositively
[4] and Hamilton
N) such that
of the isoenergy
curved
we have an extension
[6]: given 4 E C3(B,
u = 4 on dB,
allows us to impose a condition
a mixture
0
map from a ball B,
we obviously
by Eells-Sampson
*
dr
N),
and u is homotopic
on u]aB, e.g. conformality.
inequality
there is a
and the isoperimetric
to 4.
Thus we
inequality
as follows.
Corollary
l},
2.5.
Suppose N is nonpositively
curved and let I?[ = {X E ILL : 1x1 <
1 = 2,3.
i) If u : B2 + N is harmonic
and ul,yB is a constant
4xArea(u(B’))
5
ii) If u : B3 -+ N is harmonic
<
i) The first inequality
speed condition
and Theorem
Length(u(dB2))2.
5
then
VoZume(u(dB))2.
is well known.
2.4.
then
Area(u(8B3)).
and u]a~ is conformal,
(nw,)3--nE(u)n-1
Proof.
5
and ulaB is conformal,
E(u)
iii) If u : B + N is harmonic
2nE(u)
speed map, then
For the second,
use the constant
JAIGYOUNG
654
CHOE
ii) A special case of iii).
iii) Let k be the length enlargement ratio of ~13~. Then
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Received
Revised
version
received
April
30, 1997
January
10, 1998
DEPARTMENT OF MATHEMATICS, SEOUL NATIONAL UNIVERSITY,SEOUL, 151-742
E-mail address:
chodmath.snu. ac . kr