A geometrical proof of the
Hartogs extension theorem (1906)
(J. M ERKER and E. P ORTEN)
Hartogs theorem. Let Ω ⋐ Cn be a bounded domain having connected boundary. If n > 2, every function holomorphic in some connected open neighborhood of ∂Ω extend holomorphically and uniquely
inside Ω:
∀f ∈ O V(∂Ω) ∃ F ∈ O Ω ∪ V(∂Ω) F =f
V(∂Ω)
• Birth of Several Complex Variables Theory.
• Hurwitz 1897 (2nd ICM): {0} in C2 is removable.
• Hartogs 1906 (Math. Ann.): special domains.
• Cauchy’s formula and method of analytic discs.
• Osgood 1929: general statement; incorrect proof.
• Brown 1936: correct monodromy arguments ?
• Kneser 1936: CR extension phenomenon.
• Martinelli 1938: multidimensional kernel.
• Fueter 1939: quaternionic integral formula.
• Martinelli 1942: complex integral formula.
• Bochner 1943: no CR functions at all !
• Fichera 1957: jump formula and CR extension.
• Ehrenpreis 1961: ∂ with compact support.
Hurwitz theorem. O(C2 \{0}) = O(C2 ), i.e. there cannot exist isolated singularities.
z2
(ζ, z2) : |ζ| = 1
0
y1
Proof. Let f be holomorphic in C2 \{0} and compute its prolongation
Z
1
f (ζ, z2)
F (z1 , z2) :=
dζ.
2πi |ζ|=1 ζ − z1
1
x1
2
For z2 6= 0, recover f ; for z2 = 0, get extension.
Compact set ∼
= bold point.
Equivalent Hartogs theorem. Let Ω ⋐ Cn be a bounded domain. If
K ⋐ Ω is a compact such that Ω\K is connected, then:
O Ω\K = O Ω .
△ Naive intuition: Ω is a patatoid and ∂Ω is a spherical shell.
Rough patatoid
V(∂Ω)
K
∂Ω
1111111111111111111111
0000000000000000000000
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
0000000000000000000000
1111111111111111111111
111111111
000000000
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
Monodromy problems:
Recipe:
Good analytic disc
Eat a part of K
Reduce K
Restart
Ω = ball of radius r1
K = ball of radius r2
Discs:
adapted radius;
enveloping K fully
3
Overlap
– Is Hartogs’ theorem true ?. . . ?
Two known rigorous true proofs:
• Martinelli kernel.
• Ehrenpreis: ∂ with compact support.
No correct geometric proof known since 1906.
Fornæss 1998: There exists a domain ΩF ⊂ C2 which is not fillable
by discs, with the constraint that all discs remain inside ΩF .
bidisc ∆2
C2\ΩF
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
nonfillable
region
ΩF
But: (Bedford, Math. Reviews): there is an obvious filling of ΩF if we
allow discs to go outside ΩF : just take vertical discs !
4
Theorem. (M.-P ORTEN, 2006) The Hartogs extension theorem can
be proved by means of a finite number of families of analytic discs.
Heuristic illustration of the general process:
rb6
rb5
create
rb4
rb3
rb2
rb1
rb3
merge
rb2
suppress
Fig. 1: Filling the domain, creating, merging and suppressing components
5
I. Preparation of a good C ∞ boundary
1/2
• ||z|| := |z1 |2 +· · ·+|zn |2
the Euclidean norm of z = (z1, . . . , zn ) ∈
n
C .
• Bn(p, δ) := ||z − p|| < δ ball of radius δ > 0.
• If E ⊂ Cn is any set, define its δ-neighborhood
Vδ (E) := ∪p∈E Bn(p, δ).
• Let Ω ⋐ Cn , n > 2, with ∂Ω connected.
• V(∂Ω) connected neighborhood of ∂Ω.
• Origin := p0 ∈ Cn , far from Ω.
• Distance function: r(z) := ||z − p0|| = ||z||.
∂Ω
M
p0
V(∂Ω)
Fig. 2: Preparing the boundary
Lemma. There exists a C ∞ connected closed and oriented hypersurface
M ⊂ Vδ1 /2 (∂Ω) such that:
(i) M bounds a unique bounded domain ΩM with Ω ⊂ ΩM ∪ V(∂Ω);
(ii) the restriction rM (z) := r(z)M of the distance function r(z) =
||z|| to M has only a finite number κ of critical points pbλ ∈ M,
1 6 λ 6 κ, located on different sphere levels, namely
r(b
p1) < · · · < r(b
pκ );
(iii) all the (2n − 1) × (2n − 1) Hessian matrices
H[rM ](b
p1), . . . , H[rM ](b
pκ) have a nonzero determinant.
6
• kλ := number of positive eigenvalues of the (symmetric) Hessian matrix H[rM ](b
pλ).
Extrinsic Morse lemma: there exist 2n real coordinates
v, x1, . . . , xkλ , y1, . . . , y2n−kλ−1 near pbλ such that:
• the sets {v(z) = cst} correspond to the spheres {r(z) = cst} near
pbλ ;
• x1, . . . , xkλ , y1, . . . , y2n−kλ−1 provide (2n − 1) local coordinates
on the hypersurface M, whose graphed equation is normalized to
be
X
X
2
yj2 .
xj −
v=
16j6kλ
16j62n−kλ−1
7
II. Unique holomorphic extension
V
V
U1
U1
U2
intersection
connected
U2
1111111111
0000000000
intersection 1111111111
0000000000
not connected1111111111
0000000000
0000000000
1111111111
Definition. Given two connected open sets U1 ⊂ Cn and U2 ⊂ Cn with
U1 ∩ U2 nonempty, we will say that O(U1) extends holomorphically to
U1 ∪ U2 if :
• the intersection U1 ∩ U2 is connected;
• there exists an open nonempty set V ⊂ U1 ∩ U2 such that for every
f1 ∈ O(U1), there exist f2 ∈ O(U2) with f2|V = f1|V .
It then follows from the principle of analytic continuation that
f1|U1 ∩U2 = f2 |U1 ∩U2 , so that the joint function F , equal to fj on Uj for
j = 1, 2, is well defined, is holomorphic in U1 ∪ U2 and extends f1,
namely F |U1 = f1.
Reduction of the Hartogs theorem to a good boundary. Suppose that
for some δ with 0 < δ 6 δ1/2 so small that Vδ (M) ≃ M × (−δ, δ) is
a thin tubular neighborhood of the good boundary M ⊂ Vδ1 /2(∂Ω) ⊂
V(∂Ω), the Hartogs theorem holds for the pair (ΩM , Vδ (M)):
O Vδ (M) = O ΩM ∪ Vδ (M) Vδ (M ) .
Then the general Hartogs extension property holds:
O V(∂Ω) = O Ω ∪ V(∂Ω) V(∂Ω) .
8
∂Ω
M
1111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111111111111111111111111
γ
V(∂Ω)
γ
p
ΩM
q
Vδ (M)
Fig. 3: Checking connectedness of ΩM ∩ V(∂Ω)
Proof. Let f ∈ O V(∂Ω) . By assumption, its restriction
to Vδ (M) ⊂
V(∂Ω) enjoys an extension Fδ ∈ O ΩM ∪ Vδ (M) . To ascertain that f
and Fδ coincide in ΩM ∩ V(∂Ω):
• check that ΩM ∩ V(∂Ω) is connected .
• deduce that
[
ΩM ∪ Vδ (M) ∩ V(∂Ω) = ΩM ∩ V(∂Ω)
Vδ (M)
is also connected.
Summary:
• M = ∂ΩM good connected boundary
• neighborhood of ∂ΩM = Vδ (M)
• Goal = construct holomorphic extension with discs.
∂Ω
M
9
III. Hartogs and Levi-Hartogs figures
Given ε ∈ R with 0 < ε << 1 and a ∈ N with 1 6 a 6 n − 1, we split
the coordinates z ∈ Cn as (z1 , . . . , za) together with (za+1 , . . . , zn ), and
we define the (n − a)-concave Hartogs figure by
n
o
Hεn−a := max |zi | < 1, max |zj | < ε
16i6a
a+16j6n
o
[n
1 − ε < max |zi | < 1, max |zj | < 1 .
16i6a
a+16j6n
y2
C2
|z2 |
1
′
∆
Aεz2 (∆)
2
Hε2−1
′
Aεz2 (∂∆)
ε
0
1 |z1 |
0
z1
1−ε
Fig. 4: Two views of the standard Hartogs figure Hε2−1 ⊂ C2
Lemma. O Hεn−a extends holomorphically to the unit polydisc
z ∈ Cn : max |zi | < 1 = ∆n.
16i6n
Proof. The green discs have their boundaries (the green bold points) inside Hεn−a and they fill in ∆2. We just use the Cauchy formula as in the
Introduction.
• Spherical shells. For r > 1 and 0 < δ << 1, the sphere S2n−1
=
r
n
{z ∈ C : ||z|| = r} of radius r is the interior (and strongly concave)
boundary component of the spherical shell domain
Srr+δ := r < ||z|| < r + δ .
x2
10
yn
Cn
Srr+δ
LHε1 ,ε2
Tp Sr2n−1
p
S2n−1
r
Fig. 5: Relevance of the Levi-Hartogs figure
• Levi-Hartogs figure: (better than the usual)
n
o
LHε1 ,ε2 :=
max |zi | < ε1 , |xn| < ε1 , −ε2 < yn < 0
16i6n−1
o
[ n
2
ε1 − (ε1) < max |zi | < ε1 , |xn | < ε1 |yn | < ε2 .
16i6n−1
Lemma. O LHε1 ,ε2 extends holomorphically to the full parallelepiped
o
n
\
LHε1 ,ε2 :=
max |zi | < ε1, |xn | < ε1, |yn | < ε2 .
16i6n−1
• Rescale and reorient:
Φp :
ε2
0
ε1 − (ε1 )2
ε1
Bnr
xn
z 7−→ p + U z,
with U ∈ SU(n, C), sending the origin 0 ∈ LHε1 ,ε2 to p and T0LHε1 ,ε2
.
to TpS2n−1
r
ε1
z
11
IV. Quantitative Hartogs-Levi extension
2
Lemma. If ε1 = c δ and ε2 =
c δ with some appropriate positive
constant c < 1, then Φp LHε1 ,ε2 is entirely contained in the shell Srr+δ .
δ2
\
Furthermore, Φp LH
ε1 ,ε2 contains a rind of thickness c r around some
region Rp ⊂ S2n−1
whose (2n − 1)-area equals ≃ c δ 2n−1 .
r
yn
cδ 2n−2
cδ
2
c δr
p
S2n−1
r
Srr+δ
xn
Srr+δ
Φp LHε1 ,ε2
cδ
Rp
z′
2
p
c δr
2
c δr
Fig. 6: Size of the piece of (green lemon) rind
• Rind of thickness η > 0 around R ⊂ S2n−1
:
r
Rind R, η := (1 + s)z : z ∈ R, |s| < η/r .
R
Rind R, η
η
η
Proposition. Let R ⊂ S2n−1
(with r > 1 and n > 2) be a relatively open
r
∞
set having C boundary N := ∂R and let δ > 0 with 0 < δ << 1. Then
holomorphic functions in the open piece of shell:
n
n
R
∪
N
:=
C
Shellr+δ
R
∪
N
B
∩
V
δ
r
r
2
do extend holomorphically to a rind of thickness c δr around R by means
of a finite number 6 C area(R)
δ 2n−1 of Levi-Hartogs figures.
R
Shellr+δ
R
∪
N
r
R
N = ∂R
N = ∂R
Sr2n−1
2
Rind R, c δ r
−1
Sr2n−1
Fig. 7: Semi-global extension from a pseudoconcave piece of shell
12
V. Filling domains outside balls of decreasing radius

0 < δ 6 δ1
neighborhood Vδ (M)





2 6 r(b
p1) < · · · < r(b
pκ )
Morse radii



smallness of δ1
δ 6 δ1 << min rbλ+1 − rbλ
16λ6κ−1




η := c δ 2 rbκ−1
uniform useful rind thickness




η << δ
thickness of extensional rinds is tiny
Starting the filling.
p
b
V
M
κ
δ
>r >r
0000000000
1111111111
11111111111
00000000000
Ω>r
0000000000
1111111111
00000000000
11111111111
0000000000
1111111111
00000000000
11111111111
0000000000
1111111111
M>r
00000000000
11111111111
0000000000
1111111111
00000000000
11111111111
0000000000
V1111111111
00000000000
11111111111
δ (M )
0000000000
1111111111
00000000000000
11111111111111
00000000000
11111111111
0000000000
1111111111
00000000000000
11111111111111
00000000000
11111111111
0000000000
1111111111
00000000000000
11111111111111
00000000000
11111111111
W
r
V
0000000000
1111111111
00000000000000
11111111111111
δ (M )
00000000000
11111111111
Cn
Ω>r
pbκ
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
rbκ
00000000000000
11111111111111
0000000000
1111111111
11111111111111
00000000000000
00000000000
11111111111
0000000000
1111111111
00000000000000
11111111111111
00000000000
11111111111
0000000000
1111111111
00000000000000
11111111111111
r
00000000000
11111111111
r
0000000000
1111111111
00000000000000
11111111111111
N
N
00000000000
11111111111
r
r
0000000000
1111111111
00000000000000
11111111111111
R
r
r
0000000000
1111111111
00000000000000
11111111111111
0000000000
1111111111
00000000000000
11111111111111
0000000000
1111111111
00000000000000
11111111111111
M
M
0000000000
1111111111
00000000000000
11111111111111
0000000000
1111111111
r
b
00000000000000
11111111111111
κ−1
0000000000
1111111111
00000000000000
11111111111111
pbκ−1
0000000000
1111111111
00000000000000
11111111111111
pbκ−1
0000000000
1111111111
00000000000000
11111111111111
00000000000000
11111111111111
00000000000000
11111111111111
M
00000000000000
11111111111111
Fig. 8: Filling the domain from the farthest point
00000000000000
11111111111111
Proposition. For every cutting radius r with rbκ−1 < r < rbκ arbitrarily
close to rbκ−1, holomorphic functions in
Vδ M>r
>r
= Vδ M>r ∩ ||z|| > r
do extend holomorphically and uniquely to Ω>r by means of a finite
2n−1 rbκ −r number 6 C rbδκ
of Levi-Hartogs figures.
η
Lemma. For every radius r′ with rbκ−1 < r < r′ < rbκ ,
′
Shellrr′ +δ Rr′ ∪ Nr′
is contained in Ω>r′
[
Vδ M>r
>r
.
13
Ω>r′
Ω>r′ −η
pbκ
pbκ
11111111111111111
00000000000000000
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
000000000000000000
111111111111111111
′
′
′ 11111111111111111
000000000000000000
111111111111111111
r +δ
r
r 00000000000000000
−
′
′
Shellr′ Rr ∪ Nr
00000000000000000
11111111111111111
000000000000000000
111111111111111111
′
,
η
R
Rind
r
00000000000000000 111111111111111111
11111111111111111
000000000000000000
00000000000000000
11111111111111111
000000000000000000
00000000000000000 111111111111111111
11111111111111111
000000000000000000
111111111111111111
00000000000000000
11111111111111111
000000000000000000
r 11111111111111111
r 111111111111111111
00000000000000000
000000000000000000
111111111111111111
ΩM
M
M
Fig. 9: A shell contained in the cap-shaped domain and the associated rind
Lemma. The following intersection is connected:
\
Rind Rr′ , η
Ω>r′ ∪ Vδ M>r >r .
Furthermore, the union contains
Ω>r′ −η ∪ Vδ M>r
>r
.
14
VI. Creating, merging and suppressing subdomains
Lemma. Fix a radius r satisfying rbλ < r < rbλ+1 for some λ with
1 6 λ 6 κ − 1. Then:
(a) Tz M + Tz S2n−1
= Tz Cn at every point z ∈ M ∩ S2n−1
;
r
r
(b) the intersection M ∩ S2n−1
is a C ∞ compact hypersurface Nr ⊂
r
S2n−1
of codimension 2 in Cn , without boundary having finitely
r
many connected components;
(c) Nr′′ is diffeomorphic to Nr′ , whenever rbλ < r′′ < r′ < rbλ+1;
(d) M>r = M ∩{||z|| > r} has finitely many connected components
c
M>r
, with 1 6 c 6 cλ , for some cλ < ∞ which is independent of
r;
c
c
bλ < r′′ < r′ < rbλ+1,
(e) M>r
′′ is diffeomorphic to M>r′ , whenever r
for all c with 1 6 c 6 cλ ;
(f) M ∩ {r′′ < ||z|| < r′ } is diffeomorphic to Nr′ × (0, 1);
Fig. 12: Possible topologies of the cut out hypersurfaces M>r
15
Main Proposition. Fix a radius r satisfying rbλ < r < rbλ+1 for some λ
c
with 1 6 λ 6 κ − 1 and let M>r
, c = 1, . . . , cλ , denote the collection of
connected components of M ∩ {||z|| > r}. Then:
c
e c which is
(i) each M>r
bounds in {||z|| > r} a unique domain Ω
>r
relatively compact in Cn ;
e c , namely:
(ii) the boundary in Cn of each Ω
>r
ec
e c = M c ∪ Nc ∪ R
∂Ω
>r
>r
r
r
c
consists of M>r
together with some appropriate union Ncr of
finitely many connected components of Nr = M ∩ {||z|| = r}
ec ⊂ S2n−1 delimited by Nc ;
and with an appropriate region R
r
r
r
c
c
1
2
e >r and Ω
e >r , associated to two different con(iii) two such domains Ω
c1
c2
nected components M>r and M>r
of M>r , are either disjoint or
one is contained in the other;
e c2 are either disjoint or one
e c1 and R
(iv) for c1 6= c2 , the regions R
r
r
is contained in the other, while their boundaries Ncr1 and Ncr2 are
always disjoint;
f holomorphic in
(v) for each c = 1, . . . , cλ , every function
c
Vδ M>r >r has a restriction to Vδ M>r >r which extends holoe c by means of a finite number of
morphically and uniquely to Ω
>r
Levi-Hartogs figures.
111111111111111
000000000000000
1010 pb
000000000000000
111111111111111
00000000000000000000000000000000
11111111111111111111111111111111
000000000000000
111111111111111
00000000000000000000000000000000
11111111111111111111111111111111
00000000000000000000000000000000
11111111111111111111111111111111
00000000000000000000000000000000
11111111111111111111111111111111
M
V
00000000000000000000000000000000
11111111111111111111111111111111
pb
00000000000000000000000000000000
11111111111111111111111111111111
1
0
V M
00000000000000000000000000000000
11111111111111111111111111111111
r
00000000000000000000000000000000
11111111111111111111111111111111
000000000000000
111111111111111
00000000000000000000000000000000
11111111111111111111111111111111
000000000000000
111111111111111
00000000000000000000000000000000111111111111111
11111111111111111111111111111111
000000000000000
λ
δ
λ+1
δ
c
−
−
>b
rλ+1
>b
rλ+1
−
rbλ+1
′
c
rλ+
>b
rλ+ >b
rbλ+
16
c
M>b
r +
v
v
λ
λ
ec
Ω
>b
r +
λ
M
2η
2η
Vδ (M)
pbλ
x
+
rbλ
Cη
η/2
η/2
η/2
η/2
Rind−
Cη
c
M>b
r +
x
pbλ
Rind−
4η 1/2
x
2η
−
rbλ
Vδ (M)
2η 1/2
rbλ
Fig. 14: The radial (pseudo)cube Cη centered at pbλ
Fig. 15: Growing of superlevel domains near a local maximum or minimum
0000000000000000000
1111111111111111111
c
11111111111111111
00000000000000000
c
0000000000000000000
1111111111111111111
Vδ M
M>b
ec
00000000000000000
11111111111111111
>b
r
>b
r
Ω
r
0000000000000000000
1111111111111111111
>b
r
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
ec
00000000000000000
11111111111111111
Ω
0000000000000000000
1111111111111111111
>b
r
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
c,1
e
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
Rrb
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
c,2
00000000000000000
11111111111111111
e
0000000000000000000
1111111111111111111
Rrb
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
+
0000000000000000000
rb1111111111111111111
00000000000000000
11111111111111111
λ
0000000000000000000
1111111111111111111
00000000000000000
11111111111111111
0000000000000000000
00000000000000000
11111111111111111
rb1111111111111111111
λ
0000000000000000000
1111111111111111111
−
00000000000000000
11111111111111111
Rind
0000000000000000000
1111111111111111111
pbλ
00000000000000000
11111111111111111
−
Rind−
e c,1
0000000000000000000
R
rb1111111111111111111
00000000000000000
11111111111111111
λ
rb
0000000000000000000
000000000000000001111111111111111111
11111111111111111
0000000000000000000
1111111111111111111
+
λ
+
λ
+
λ
+
λ
+
λ
+
λ
+
λ
+
λ
Fig. 16: Two distinct Hartogs-Levi fillings at a point of Morse coindex 2n − 1
17
2
v = x2 − y12 − · · · − y2n−1
Cη
−
M>ε
c1
M>b
r+
v
λ
+
M>ε
{v = ε}
e c1+,1
R
r
b
λ
e c2+,1
R
r
b
λ
λ
R−
ε
x
pbλ
M
c2
M>b
r+
Cη
rbλ+
pbλ
rbλ−
e ∗−
R
r
b
λ
y
Fig. 17: Slices and superlevel sets at a Morse point of coindex 1
0000000000
1111111111
η
v
=
−
11111
00000
00000
11111
000000000
111111111
v= η
0000000000
1111111111
000
111
00
11
0
1
v11111
= 0111
000
00
11
000
111
00
11
00000
11111
00000
000000000
111111111
v= η
0000000000
1111111111
000
111
00
11
0000
10101111
0000
1111
0
1
0
1
000
111
00
11
00
11
000
111
00
11
000
111
00
11
00000
11111
00000
11111
000000000
111111111
00
11
0000000000
1111111111
000
111
00
11
0000
1111
0000
1111
0
1
00
11
0
1
000
111
00
11
00
11
00
11
000
111
000
111
00
11
000
111
00
11
00000
11111
00000
11111
000000000
111111111
00
11
0000000000
1111111111
000
111
00
11
0000
1111
0000
1111
0
1
00
11
0
1
000
111
00 10101111
11
00
11
00
11
000
111
000
111
00
11
000
111
00
11
00000
11111
00000
11111
000000000
111111111
0
1
100
0
0000000000
1111111111
000
111
00
11
0000
0000
1111
0
1
00
11
11
000
111
0
1
00
11
000
111
00
11
00000
11111
00000
11111
000000000
111111111
merge
0
1
0000000000
1111111111
000
00
11
0000 1111
101111
0000
0 11111
1
00
11
0111
1
00
11
000
111
00
11
00000
00000
11111
000000000
111111111
0000000000
1111111111
000
111
00
11
Fig. 18: Sliced view of the merging of the two domains in case (a)
000
111
000000000
111111111
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
substract
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
000
111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
00
11
00
11
00
11
Fig. 19: Sliced view of the substraction of the left domain in case (b)
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
000000000
111111111
0000000000
1111111111
0000000000
1111111111
000000000
111111111
0000000001111111111
111111111
00000000001111111111
0000000000
Fig. 20: Sliced view of the growing of the external domain in case (e)
2
3
1
2
1
2
18
VII. Hartogs extension on singular complex spaces
covered:
• X, OX complex space openly
S
j∈J Xj = X.
• Holomorphic isomorphisms :
ϕj : Xj −→ Aj := complex analytic subset of
ej := ball ⊂ CNj , some Nj > 1
B
• A C ∞ function : ρ : X → R is called exhaustion if sublevel sets
{ρ < c} are ⊂⊂ X for every c ∈ R.
ej → R.
• Locally: ρXj = ρej ◦ ϕj for some C ∞ function ρej = B
• Definition : ρ is strongly (n − 1)-convex if the extension ρej can be
chosen so that:
Levi form(e
ρj ) has at least Nj − n + 2 eigenvalues > 0
⇒ ρXreg has at least 2 eigenvalues > 0
⇒ boundaries of super level sets (when smooth)
{ρ > c} ∩ Xreg have at least 1 eigenvalue < 0
• Observation: This is the weakest assumption to insure pseudoconcavity, in the category of “strong” Levi-form assumptions.
• Open question: How to sharpen such assumptions ?
• Definition : X is strongly (n − 1)-complete if X possesses a strongly
(n − 1)-convex exhaustion function.
19
• Smooth case: X = Xreg .
Theorem. (Andreotti-Hill 1972) If X = Xreg is (n − 1) complete of
dimC X > 2, then for every domain Ω ⊂ X and every compact K ⊂ Ω
such that Ω\K is connected:
O(Ω\K) = O(Ω)
Ω\K
• Proof (without ∂) :
• Similarly as previously.
• Morse combinatorics of connected components of {ρ > r} ∩ M.
• But: (cf. bump method) must take account of:
Crit(ρ) := p ∈ Ω : dρ(p) = 0
• Arrange in advance that: Crit(ρ) ∩ M = ∅.
• Since LF(ρ) has > 2 eigenvalues > 0, in Morse coordinates at a critical
point where
2
ρ = x21 + · · · + x2k − y12 − · · · − y2n−k
,
we have k > 2, hence all sup-level sets
are locally connected.
ρ>r
20
ρ < c̃
ρ > c̃
p̂
ρ = c̃
ρ = c̃ − ǫ
M
E XTENSION THROUGH SINGULAR LEVELS
• Observation: Extending from shells
{z : a < ρ(z) < b}
into complete sublevel sets {ρ < b} is much easier, because the combinatorics of connected components of {ρ > r} ∩ M disappears.
ρ = ρ(p̂)
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
y1, . . . , y2n−k
x1 , . . . , xk
ρ > ρ(p̂)
ρe = ρ(p̂)
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
11
00
111p̂
000
ρ < ρ(p̂)
E XTENSION THROUGH CRITICAL LEVELS
ρ > ρ(p̂)
21
• Singular case: Assume from now on that X is normal.
• Otherwise: There exist Stein surfaces S with 1 isolated singularity p
such that holomorphic functions on Sreg fail to extend near p.
• Riemann removability theorem: locally bounded holomorphic functions defined on Xreg extend through Xsing , hence to all of X.
• Levi meromorphic theorem: Meromorphic functions on Xreg extend
meromorphically to X.
Theorem. (M. P ORTEN 2007) If X is a normal (n−1)-complete space
of pure dimension n > 2, then for every domain Ω ⊂ X and every
compact K ⊂ Ω such that Ω\K is connected:
O(Ω\K) = O(Ω)
,
Ω\K
and similarly for meromorphic functions.
Main Proposition. Let X, Ω, K be as above.
(i) Every meromorphic function f defined on Ω\K reg has a unique
meromorphic extension F to Ωreg.
(ii) If f is holomorphic on Ω\K reg , then F is holomorphic on Ωreg
and:
sup |F | = sup |f |.
Ωreg
[Ω\K]reg
Lemma. (Demailly 1990) There exists a quasi-psh function v on X such
that
Xsing = v = −∞
• Quasi-psh = psh + C ∞ or more precisely, the local extensions vej
ej satisfy:
defined on B
vj = u
e
ej + rj
with a psh u
ej and a C ∞ rj .