Invent math (2013) 193:149–158
DOI 10.1007/s00222-012-0423-2
Suita conjecture and the Ohsawa-Takegoshi
extension theorem
Zbigniew Błocki
Received: 3 April 2012 / Accepted: 26 August 2012 / Published online: 8 September 2012
© The Author(s) 2012. This article is published with open access at Springerlink.com
Abstract We prove a conjecture of N. Suita which says that for any bounded
2 ≤ πK , where c (z) is the logarithmic capacity
domain D in C one has cD
D
D
of C \ D with respect to z ∈ D and KD the Bergman kernel on the diagonal.
We also obtain optimal constant in the Ohsawa-Takegoshi extension theorem.
1 Introduction
Suita [16] (see also [15, p. 179]) conjectured that for a bounded domain D in
C one has
2
cD
≤ πKD .
Here
(1)
cD (z) = exp lim GD (ζ, z) − log |ζ − z|
ζ →z
is the logarithmic capacity of the complement of D with respect to z ∈ D,
where GD (·, z) the (negative) Green function for D with pole at z, and
2
2
|f | dλ ≤ 1
KD (z) = sup f (z) : f holomorphic in D,
D
Z. Błocki ()
Instytut Matematyki, Uniwersytet Jagielloński, Łojasiewicza 6, 30-348 Kraków, Poland
e-mail: [email protected]
Z. Błocki
e-mail: [email protected]
150
Z. Błocki
denotes the Bergman kernel on the diagonal. One can easily show that we
have equality in (1) if D is simply connected. Suita in [16], using the theory
of elliptic functions, proved strict inequality in (1) when D is an annulus, and
hence also any regular doubly connected domain.
This conjecture has geometric interpretation and consequences. Suita also
observed that
KD =
1 ∂2
(log cD )
π ∂z∂ z̄
and thus that (1) is equivalent to the fact that the curvature of the invariant
metric cD |dz| is bounded above by −4. Therefore (1) means precisely that
maximal curvature is attained on the boundary (if D is smooth then one can
show that this curvature is equal to −4 on ∂D, or equivalently that the ratio
2 /πK is equal to 1 there). This is not the case for other invariant metcD
D
rics related to the Bergman kernel, e.g. KD |dz|2 and the Bergman metric
(log KD )zz̄ |dz|2 (see [9] and [19]). On the other hand, this property is satisfied for the Carathéodory metric (see [17] and [18]) but this is much simpler
to prove.
The relation of the Suita conjecture to the extension theorem was first observed and explored by Ohsawa [12]: to prove (1) for a given z ∈ D we have
to find a holomorphic f in D with f (z) = 1 and
|f |2 dλ ≤
D
π
.
(cD (z))2
Using methods developed in [14] he proved the estimate
2
≤ C πKD
cD
with C = 750. This was later improved to C = 2 in [5], where the main tool
was an estimate from [1]. A slightly better constant C = 1.954 was recently
obtained in [10], see also [6] for a much simpler proof.
Our goal is to show the following version of the Ohsawa-Takegoshi theorem with optimal constant which in particular implies the Suita conjecture:
Theorem 1 Assume that Ω ⊂ Cn−1 × D is pseudoconvex, where D is a
bounded domain in C containing the origin. Then for any holomorphic f
in Ω := Ω ∩ {zn = 0} and ϕ plurisubharmonic in Ω one can find a holomorphic extension F of f to Ω with
π
|F |2 e−ϕ dλ ≤
|f |2 e−ϕ dλ .
2
(c
(0))
D
Ω
Ω
Suita conjecture
151
This kind of result, with a worse constant though, was first formulated in
[8] (it used the proof of the Ohsawa-Takegoshi theorem from [2]), see also
[13].
Our proof is in big part motivated by the recent work of Chen [7] who
showed that the Ohsawa-Takegoshi theorem can deduced directly from the
¯
classical Hörmander estimate for the ∂-equation
[11]. We use Chen’s idea in
¯
the proof of Theorem 2 below. It improves a related ∂-estimate
from [6] which
¯
gave Theorem 1 but only with the same constant as in [10]. This ∂-estimate
was one of the crucial improvements compared to [6]. Another one was the
solution of the following ODE problem: find g ∈ C 0,1 (R+ ), h ∈ C 1,1 (R+ )
such that h is convex and decreasing, g + log t, h + log t vanish at ∞, and
(g )2 2g−h+t
1 − e
≥ 1.
h
The author would like to express his gratitude to Takeo Ohsawa for his
constant interest and encouragement throughout the years and to Peter Pflug
for essentially introducing him to the theory of Bergman kernel back in 1998.
2 The proof
Similarly as in [7] our main tool will be the Hörmander estimate: if Ω is
pseudoconvex in Cn and
α=
αj d z̄j ∈ L2loc,(0,1) (Ω)
j
¯ = 0 then for any smooth, strongly plurisubharmonic ϕ in Ω
is such that ∂α
one can find u ∈ L2loc (Ω) such that
¯ =α
∂u
and
2 −ϕ
|u| e
dλ ≤
Ω
Ω
Here
|α|2i∂ ∂ϕ
¯ =
−ϕ
|α|2i∂ ∂ϕ
¯ e dλ.
(2)
ϕ j k̄ ᾱj αk
j,k
(where (ϕ j k̄ ) is the inverse transposed of the complex Hessian (∂ 2 ϕ/∂zj ∂ z̄k ))
¯
denotes the length of α with respect to the Kähler metric i∂ ∂ϕ.
152
Z. Błocki
It was observed in [4] that (2) makes also sense (and indeed remains
true) for arbitrary plurisubharmonic ϕ: instead of |α|2i∂ ∂ϕ
¯ one should take any
(Ω)
with
H ∈ L∞
loc
¯
i ᾱ ∧ α ≤ H i∂ ∂ϕ.
¯
In this convention we formulate our new estimate for the ∂-equation:
¯
Theorem 2 Let α ∈ L2loc,(0,1) (Ω) be a ∂-closed
form in a pseudoconvex do1,2
(Ω) is
main Ω in Cn . Assume that ϕ is plurisubharmonic in Ω, ψ ∈ Wloc
2
¯
locally bounded from above and they satisfy |∂ψ|i∂ ∂ϕ
¯ ≤ H < 1 in Ω for some
∞
2
¯ = α and such that
H ∈ Lloc (Ω). Then there exists u ∈ Lloc (Ω) solving ∂u
for every b > 0
1 + bH 2ψ−ϕ
1
e
|u|2 (1 − H )e2ψ−ϕ dλ ≤ 1 +
|α|2i∂ ∂ϕ
dλ.
¯
b
1−H
Ω
Ω
If in addition H ≤ δ < 1 on supp α then
|u| (1 − H )e
2
2ψ−ϕ
Ω
√ δ
dλ ≤
|α|2 ¯ e2ψ−ϕ dλ.
√
1 − δ Ω i∂ ∂ϕ
1+
(3)
Proof By standard approximation we may assume that ϕ is smooth up to the
boundary, strongly plurisubharmonic, and that ψ is bounded in Ω. Note that
then L2 (Ω, eψ−ϕ ), L2 (Ω, e−ϕ ) and L2 (Ω) are the same sets. We use a trick
¯ = α in L2 (Ω, eψ−ϕ ) which
from [3]: define u to be the minimal solution to ∂u
means that u is perpendicular to ker ∂¯ there. Then v := ueψ is perpendicular
¯ = β, where
to ker ∂¯ in L2 (Ω, e−ϕ ) and thus it is the minimal solution to ∂v
¯ ∈ L2
β := eψ (α + u ∂ψ)
loc,(0,1) (Ω).
By Hörmander’s estimate (2)
−ϕ
|v|2 e−ϕ dλ ≤
|β|2i∂ ∂ϕ
¯ e dλ
Ω
Ω
which means that
¯ 2 ¯ e2ψ−ϕ dλ
|u|2 e2ψ−ϕ dλ ≤
|α + u ∂ψ|
i∂ ∂ϕ
Ω
Ω
≤
Ω
√
2ψ−ϕ
2
|α|2i∂ ∂ϕ
dλ.
¯ + |u| H e
¯ + 2|u| H |α|i∂ ∂ϕ
Suita conjecture
153
For t > 0 we will get
|u|2 (1 − H )e2ψ−ϕ dλ
Ω
2
−1 H
2
+ t|u| (1 − H ) e2ψ−ϕ dλ.
|α|i∂ ∂ϕ
1+t
≤
¯
1
−
H
Ω
We obtain the first estimate if we take t := 1/(b + 1), whereas the second one
follows if we let b := δ −1/2 .
Proof of Theorem 1 Using appropriate approximation we may assume that
Ω is bounded, smooth and strongly pseudoconvex, ϕ is smooth up to the
boundary and that f is holomorphic in a neighborhood of Ω . Fix ε > 0 and
define
f (z )χ (−2 log |zn |)
α := ∂¯ f z χ −2 log |zn | = −
d z̄n ,
z̄n
where we write z = (z , zn ) and χ ∈ C 0,1 (R) is non-decreasing and such that
χ = 0 on {t ≤ −2 log ε}, χ (∞) = 1 (it will be precisely determined later).
Note that α is defined in Ω for sufficiently small ε, supp α ⊂ {|zn | ≤ ε}, and
¯ = α the function
for any solution of ∂u
(4)
F := f z χ −2 log |zn | − u
is a holomorphic extension of f provided that u = 0 on Ω .
In order to use Theorem 2 and define appropriate weights ϕ , ψ the choice
of the following two functions on R+ is absolutely crucial:
h(t) := − log t + e−t − 1 ,
g(t) := − log t + e−t − 1 + log 1 − e−t .
They solve the ODE problem formulated in the introduction: one can check
that they are decreasing, convex,
(g )2 2g−h+t
=1
(5)
1 − e
h
and
lim h(t) − 2g(t) + log −h (t) = 0.
t→∞
(6)
(In fact, as noticed by the referee, we have g = log(−h ). This substitution
reduces (5) to
d 2 −h e
= e−t
dt 2
154
Z. Błocki
and enables to solve the desired ODE problem immediately.)
For G := GD (·, 0) we can write G = log |ζ | + v, where v is a bounded
harmonic function in D. There are positive constants C0 , C1 such that
2G − 2 log |ζ | ≤ C0
and
in D
2Gζ − 1 ≤ C1 near the origin.
ζ
For M := −2 log ε − C0 we define
h(t),
η(t) :=
−δ log(t − M + a) + b,
and
t < M,
t ≥M
g(t),
t < M,
γ (t) :=
−δ log(t − M + a) + b, t ≥ M,
where δ = M −1/2 and a, b, b are chosen in such a way that η ∈ C 1,1 (R+ ) and
γ ∈ C 0,1 (R+ ). Then
a = a(ε) = −
δ
h (M)
,
b = b(ε) = h(M) + δ log a,
b =
b(ε) = g(M) + δ log a.
Note that δ is selected so that
lim δ(ε) = 0,
ε→0
lim a(ε) = ∞.
ε→0
(7)
Now we are ready to define
ϕ := ϕ + 2G + η(−2G),
ψ := γ (−2G).
1,2
(D) is locally bounded from
Then ϕ is plurisubharmonic in Ω and ψ ∈ Wloc
above. We have
(−2G))2
(γ
< 1 in Ω,
2
2
¯
¯
|∂ψ|
= ¯ ϕ ≤ |∂ψ|i∂ ∂(η(−2G))
¯
i∂ ∂
η (−2G) = δ on supp α
Suita conjecture
155
(note that supp α ⊂ {−2G ≥ M}) and
2
|α|2i∂ ∂
=
¯ ϕ ≤ |α|i∂ ∂(η(−2G))
¯
|f (z )|2 (χ (−2 log |zn |))2
,
4|Gzn |2 |zn |2 η (−2G)
where we follow the convention discussed before Theorem 2. Since the function −δ log(t − M + a) + t is increasing in t by (5) we have
(γ )2 2γ −η+t = 1
on {t < M},
1 − e
2g(M)−h(M)+M
η
≥ (1 − δ)e
on {t ≥ M}
and thus for sufficiently small ε
(γ )2 2γ −η+t
≥1
1 − e
η
¯ = α such that
on R+ . Theorem 2 now gives a solution u = uε of ∂u
√
2ψ−
δ
1
+
2 −ϕ
2
2
ϕ
¯
|u| e dλ ≤
|u| 1 − |∂ψ|
dλ ≤
√ A(ε),
¯ϕ e
i∂ ∂
1− δ
Ω
Ω
where
A(ε) :=
Ω
|f (z )|2 (χ (−2 log |zn |))2 2ψ−
ϕ
e
dλ.
4|Gzn |2 |zn |2 η (−2G)
We have
lim A(ε) ≤ lim B (ε)
ε→0
ε→0
Ω
|f |2 e−ϕ dλ
where
B (ε) :=
(χ (−2 log |ζ |))2 2ψ−η(−2G)−2G
e
dλ(ζ ).
2
2 {|ζ |≤ε} 4|Gζ | |ζ | η (−2G)
We now want to replace G by log |ζ | in the above integral. Note that
lim sup |ζ |2 e−2G =
ε→0 {|ζ |≤ε}
and near the origin
1
(cD (0))2
2
4|Gζ |2 |ζ |2 ≥ 1 − C1 |ζ | .
Moreover on {|ζ | ≤ ε} we have
2ψ − η(−2G) ≤ 2γ −2 log |ζ | − C0 − η −2 log |ζ | − C0
(8)
156
Z. Błocki
C0
≤ 2γ −2 log |ζ | − η −2 log |ζ | + δ log 1 +
a
and
η (−2G) ≥ η −2 log |ζ | + C0
a + C0
≥
a + 2C0
2
η −2 log |ζ | .
Hence, from (7) it follows that
lim B (ε) ≤
ε→0
1
(ε),
lim B
(cD (0))2 ε→0
where
(χ (−2 log |ζ |))2 2γ (−2 log |ζ |)−η(−2 log |ζ |)
dλ(ζ )
e
2 {|ζ |≤ε} |ζ | η (−2 log |ζ |)
∞
(χ )2 e2γ −η
=π
dt.
η
M+C0
(ε) =
B
By the Schwarz inequality the optimal choice of increasing χ with
∞
χ (∞) =
χ dt = 1
M+C0
is
0,
χ (t) := 1 t
where w = η eη−2γ and
t < M + C0 ,
c M+C0 w ds, t ≥ M
∞
c = M+C0 w dt. Then
+ C0 ,
π
η−2γ dt
M+C0 η e
(ε) = ∞
B
and
∞
η−2γ
η e
M+C0
dt = δe
b−2
b
∞
(t + a)δ−2 dt
C0
C0
1
1+
=
1−δ
a
δ−1
By (6) and (7)
(ε) = π
lim B
ε→0
eh(M)−2g(M)+log(−h (M)) .
Suita conjecture
157
and we have thus obtained that
lim
|uε |2 e−ϕ dλ ≤
ε→0 Ω
π
(cD (0))2
Ω
|f |2 e−ϕ dλ .
Finally, we note that if 0 <
ε ≤ ε then
(γ (−2G))2 2ψ−η(−2G)−2G
1 − e
dλ(ζ )
η (−2G)
{|ζ |≤
ε}
e2γ (−2 log |ζ |)−η(−2 log |ζ |)
1
dλ(ζ )
≥
C(ε) {|ζ |≤ε}
|ζ |2
πe2b−b ∞
(t − M + a)−δ dt
=
C(ε) −2 logε
=∞
and (8) implies that u = 0 almost everywhere on Ω . The weak limit of
F = Fε given by (4) is the required extension.
Open Access This article is distributed under the terms of the Creative Commons Attribution
License which permits any use, distribution, and reproduction in any medium, provided the
original author(s) and the source are credited.
References
1. Berndtsson, B.: Weighted estimates for ∂¯ in domains in C. Duke Math. J. 66, 239–255
(1992)
2. Berndtsson, B.: The extension theorem of Ohsawa-Takegoshi and the theorem of
Donnelly-Fefferman. Ann. Inst. Fourier 46, 1083–1094 (1996)
3. Berndtsson, B.: Weighted estimates for the ∂-equation. In: Complex Analysis and Geometry, Columbus, OH, 1999. Ohio State Univ. Math. Res. Inst. Publ., vol. 9, pp. 43–57.
de Gruyter, Berlin (2001)
4. Błocki, Z.: The Bergman metric and the pluricomplex Green function. Trans. Am. Math.
Soc. 357, 2613–2625 (2005)
5. Błocki, Z.: Some estimates for the Bergman kernel and metric in terms of logarithmic
capacity. Nagoya Math. J. 185, 143–150 (2007)
6. Błocki, Z.: On the Ohsawa-Takegoshi extension theorem. http://gamma.im.uj.edu.pl/
~blocki (2012)
7. Chen, B.-Y.: A simple proof of the Ohsawa-Takegoshi extension theorem. arXiv:1105.
2430v1
8. Dinew, Ż.: The Ohsawa-Takegoshi extension theorem on some unbounded sets. Nagoya
Math. J. 188, 19–30 (2007)
9. Dinew, Ż.: An example for the holomorphic sectional curvature of the Bergman metric.
Ann. Pol. Math. 98, 147–167 (2010)
10. Guan, Q., Zhou, X., Zhu, L.: On the Ohsawa-Takegoshi L2 extension theorem and the
Bochner-Kodaira identity with non-smooth twist factor. J. Math. Pures Appl. 97, 579–601
(2012)
158
Z. Błocki
11. Hörmander, L.: L2 estimates and existence theorems for the ∂¯ operator. Acta Math. 113,
89–152 (1965)
12. Ohsawa, T.: Addendum to “On the Bergman kernel of hyperconvex domains”. Nagoya
Math. J. 137, 145–148 (1995)
13. Ohsawa, T.: On the extension of L2 holomorphic functions V—effects of generalization.
Nagoya Math. J. 161, 1–21 (2001)
14. Ohsawa, T., Takegoshi, K.: On the extension of L2 holomorphic functions. Math. Z. 195,
197–204 (1987)
15. Sario, L., Oikawa, K.: Capacity Functions. Grundlehren Math. Wiss., vol. 149. Springer,
Berlin (1969)
16. Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46,
212–217 (1972)
17. Suita, N.: On a metric induced by analytic capacity. Kodai Math. Semin. Rep. 25, 215–218
(1973)
18. Wong, B.: On the holomorphic curvature of some intrinsic metrics. Proc. Am. Math. Soc.
65, 57–61 (1977)
19. Zwonek, W.: Asymptotic behavior of the sectional curvature of the Bergman metric for
annuli. Ann. Pol. Math. 98, 291–299 (2010)