• Nie Znaleziono Wyników

Completeness of the Bergman metric on non-smooth pseudoconvex domains

N/A
N/A
Protected

Academic year: 2021

Share "Completeness of the Bergman metric on non-smooth pseudoconvex domains"

Copied!
11
0
0

Pełen tekst

(1)

POLONICI MATHEMATICI LXXI.3 (1999)

Completeness of the Bergman metric on non-smooth pseudoconvex domains

by Bo-Yong Chen (Shanghai)

Abstract. We prove that the Bergman metric on domains satisfying condition S is complete. This implies that any bounded pseudoconvex domain with Lipschitz boundary is complete with respect to the Bergman metric. We also show that bounded hyperconvex domains in the plane and convex domains in C

n

are Bergman comlete.

1. Introduction. Let D ⊂ C

n

be a bounded domain and let K

D

(z, w) be the Bergman kernel. The Bergman metric on D is defined as follows:

ds

2D

= X

n j,k=1

2

log K

D

(z, z)

∂z

j

∂z

k

dz

j

dz

k

. In [11] Kobayashi posed an interesting question:

Which bounded pseudoconvex domains in C

n

are complete with respect to the Bergman metric?

The assumption of pseudoconvexity is necessary (cf. [4]). It is quite clear that any bounded pseudoconvex domain with C

-boundary is complete w.r.t. ds

2D

(cf. [14]). In [13] Ohsawa has proved that the Bergman metric of any pseudoconvex domain with C

1

-boundary is complete. In this article, we first study a class of pseudoconvex domains defined as follows:

Definition. We say that a domain D in C

n

satisfies condition S if there exists a sequence {D

j

} of pseudoconvex domains with D ⊂⊂ D

j

such that

(1) Λ

j

= sup

z∈∂D

d

Dj

(z) → 0 as j → ∞, where d

Dj

(z) = d(z, ∂D

j

) is the Euclidian distance from z to ∂D

j

;

(2) there exist reals r ≥ 1 and 0 < α ≤ 1 such that Λ

j

≤ rλ

αj

, where λ

j

= inf

z∈∂D

d

Dj

(z).

1991 Mathematics Subject Classification: Primary 32H10.

Key words and phrases : Bergman metric.

[241]

(2)

We say that D satisfies condition S locally if for each z

0

∈ ∂D, there exists a ball B(z

0

, r

0

) such that D ∩ B(z

0

, r

0

) satisfies condition S.

One can easily conclude that D is pseudoconvex if D satisfies condition S locally.

We have the following main result:

Theorem 1.1. Let D be a domain which satisfies condition S locally.

Then D is complete w.r.t. the Bergman metric.

From Theorem 1.1 we can deduce

Theorem 1.2. The Bergman metric of any bounded pseudoconvex do- main with Lipschitz boundary is complete.

It is known from [5] that any bounded pseudoconvex domain with Lip- schitz boundary is hyperconvex, that is, it admits a bounded continuous plurisubharmonic (psh) exhaustion function. Naturally, one would ask:

(a) Is any bounded hyperconvex domain in C

n

complete w.r.t. the Berg- man metric?

(b) Are the hyperconvexity and the Bergman completeness equivalent?

In Section 5 we will construct a domain which is not hyperconvex but complete w.r.t. the Bergman metric. Question (a) seems to be quite difficult.

However, we can prove that the answer is affirmative for some special cases.

Given a domain D ⊂ C

n

and ζ ∈ D, we consider the function u

D

(ζ, z) = u

ζ

(z) = sup{v(z) | v is psh on D,

v < 0 and v(w) ≤ log |w − ζ| + O(1), w ∈ D}, which is the pluricomplex Green function on D with logarithmic pole at ζ (cf. [1], [5]). We obtain another main result:

Theorem 1.3. Let D be a bounded hyperconvex domain in C

n

and sup- pose that u

D

(ζ, z) is symmetric. Then the Bergman metric on D is complete.

Since the complex Green function of hyperconvex domains in C is sym- metric (cf. [12]), Theorem 1.3 implies

Corollary 1.4. Any bounded hyperconvex domain in C is complete w.r.t. the Bergman metric.

It is known from [3] that any bounded convex domain in C

n

is hyper- convex, and the pluricomplex Green function is symmetric (cf. [15]). Imme- diately we obtain

Corollary 1.5. The Bergman metric of any bounded convex domain in

C

n

is complete.

(3)

The proofs of the theorems are based on the techniques of L

2

-estimates for the ∂-equation on complete K¨ ahler manifolds due to Diederich and Oh- sawa.

2. Preliminaries. Let X be a complete manifold of dimension n and ds

2

be a Hermitian metric on X. Let C

0p,q

(X) be the set of C

-differentiable (p, q)-forms on X with compact support, and let ϕ : X → R be a continuous function. We define an inner product by

(u, v)

ds2

:=

\

X

e

−ϕ

u ∧ ∗

ds2

v

for u, v ∈ C

0p,q

(X). Here ∗

ds2

denotes the Hodge star operator associated with ds

2

. Put

kuk

ds2

:= (u, u)

1/2ds2

and denote by L

p,q

(X, ds

2

, ϕ) the space of all square integrable (p, q)-forms on X, i.e., the completion of C

0p,q

(X) with respect to the norm k · k

ds2

. Now we recall the following useful result:

Proposition 2.1 (cf. Theorem 3 in [6]). Let X be a complex manifold that admits a complete K¨ ahler metric with a positive C

global potential function η, and let ψ : X → R be another C

strictly psh function on X satisfying the estimate ∂∂ψ ≥ ∂ψ∂ψ. Furthermore, let ϕ be any C

psh function on X.

Then, for any ∂-closed (n, 1)-form g on X satisfying kgk

∂∂ψ,ϕ

< ∞, there is a measurable (n, 0)-form h satisfying ∂h = g and khk

ϕ

≤ Ckgk

∂∂ψ,ϕ

, where C is a numerical constant (independent of X, ψ, ϕ, g) and khk

2ϕ

=

|

T

X

e

−ϕ

h ∧ h|.

3. Proofs of Theorems 1.1–1.2. Let D be a bounded domain in C

n

. We denote by H

2

(D) the space of all square integrable holomorphic func- tions on D and by H(D) the space of all functions holomorphic in a neigh- bourhood of D. The L

2

-norm is denoted by k · k

D

, and kf k

D,ϕ

means the L

2

-norm of f with weight e

−ϕ

, where ϕ is a continuous real function on D.

We claim

Lemma 3.1. Let D be a domain satisfying condition S. Then H(D) is dense in H

2

(D).

P r o o f. Let f ∈ H

2

(D). Without loss of generality we can assume kf k

D

≤ 1. We define D

j,t

= {z ∈ D

j

| d

Dj

(z) > t}, t > 0. Then D

j,t

is pseudoconvex since − log d

Dj

(z)/t is a psh exhaustion function on D

j,t

, and we have

D

j,Λj

⊂ D ⊂⊂ D

j,t

for all t < λ

j

.

(4)

Put

β =

 

1/2, α = 1, 1 − α

1 − α/2 , 0 < α < 1.

Let χ : R → [0, 1] be a C

function satisfying χ|

(−∞,1+1/(2 log β))

= 1 and χ|

[1,∞)

= 0. Since each continuous psh function can be approximated by a decreasing sequence of C

strictly psh functions, there exist for each j a number ν

j

with λ

j

/2 < ν

j

< λ

j

and a C

strictly psh function u

j

defined on D

j,νj

such that

(1) − log d

Dj

(z)

λ

j

/2 < u

j

(z) < 0 on D

j,νj

, (2) u

j

(z) + log d

Dj

(z)

λ

j

/2 < (β

−1

− β

−1/2

) log Λ

j

λ

j

/2 , ∀z ∈ D.

We write e D

j

= D

j,νj

and put ψ

j

= − log(−u

j

), ̺

j

= χ



ψ

j

+ log log Λ

j

λ

j

/2 + 1

 . Then ψ

j

is a C

strictly psh function on e D

j

satisfying

∂∂ψ

j

≥ ∂ψ

j

∂ψ

j

. This gives

|∂̺

j

|

∂∂ψ

j

≤ sup |χ

|, where |∂̺

j

|

∂∂ψ

j

is the point-norm of ∂̺

j

w.r.t. the metric ∂∂ψ

j

. From (1) we obtain

supp ̺

j

⊂ D

j,Λj

⊂ D.

So we can define a C

∂-closed (n, 1)-form on e D

j

as follows:

g

j

= ∂(̺

j

f ) ∧ dz

1

∧ . . . ∧ dz

n

. Since

supp ∂̺

j

 z ∈ D

u

j

(z) > −β

−1/2

log Λ

j

λ

j

/2



 z ∈ D

− log

d

Dj

(z)

λ

j

/2 > −β

−1

log Λ

j

λ

j

/2



= D \ D

j,µj

, where

µ

j

=  2rΛ

j

, α = 1,

(2

α

r)

1/(2(1−α))

Λ

1/2j

, 0 < α < 1,

(5)

we obtain kg

j

k

2∂∂ψ

j,0

= 2

n

\

e

Dj

|∂̺

j

|

2∂∂ψ

j

|f |

2

dV

n

≤ 2

n

sup |χ

|

2

kf k

2D\D

j,µj

, where dV

k

denotes the 2k-dimensional Lebesgue measure.

Since any pseudoconvex domain admits a complete K¨ ahler metric with a positive global potential (cf. [7], p. 49), by Proposition 2.1 there exists an (n, 0)-form h

j

= e h

j

dz

1

∧ . . . ∧ dz

n

on e D

j

satisfying ∂h

j

= g

j

and

ke h

j

ke

Dj

≤ C sup |χ

| · kf k

D\Dj,µj

, where C is a numerical constant (independent of f, j, α, D).

Hence f

j

= ̺

j

f − e h

j

is holomorphic on e D

j

and satisfies

kf

j

− f k

D

≤ k(1 − ̺

j

)f k

D

+ ke h

j

k

D

≤ (1 + C) sup |χ

| · kf k

D\Dj,µj

. Given any ε > 0, there exists a δ = δ(ε) > 0 such that

kf k

D\Dδ

< ε,

where D

t

= {z ∈ D | d

D

(z) > t}, t > 0. Since Λ

j

→ 0 as j → ∞, one has µ

j

< δ/2 for all sufficiently large j. This gives D

j,µj

⊃ D

j,δ/2

for all sufficiently large j. Since d

Dj

(z) → d

D

(z) on D, there exists a j(ε) such that

D

j(ε),µj(ε)

⊃ D

δ

. Thus

kf

j(ε)

− f k

D

≤ (1 + C)ε sup |χ

|.

The proof is complete.

Lemma 3.2. lim

z→∂D

K

D

(z, z) = ∞ if D is a domain which satisfies condition S locally.

P r o o f. We can easily conclude that D satisfies the cone condition in the sense of Pflug (cf. [14], p. 399), hence we have lim

z→∂D

K

D

(z, z) = ∞.

The following localization lemma for the Bergman metric is perhaps known:

Lemma 3.3. Let D be a bounded pseudoconvex domain in C

n

and z

0

be any point in ∂D. Suppose that U , V are two open neighbourhoods of z

0

with V ⊂⊂ U . Then

ds

2D

(ζ; X) ≥ Cds

2D∩U

(ζ; X), ∀X ∈ T

1,0

(C

n

), ζ ∈ V ∩ D, where C is a constant (independent of ζ, X).

P r o o f. From the original work of Bergman (cf. [2]) we know that for any X ∈ T

1,0

(C

n

) and ζ ∈ D we have

ds

2D

(ζ; X) = K

D−1

(ζ, ζ) sup{|Xf |

2

| f ∈ H

2

(D), kf k

D

≤ 1, f (ζ) = 0}.

(6)

We choose a C

function η : C

n

→ [0, 1] such that η = 1 on a neighbourhood of V , η = 0 on C

n

\ U . Fix any point ζ ∈ D ∩ V and for any f ∈ H

2

(D ∩ U ) with kf k

D∩U

≤ 1 and f (ζ) = 0 put v = ∂(ηf ). Thus we have defined a C

∂-closed (0, 1)-form on D. Let ψ(z) = 2(n + 1) log |z − ζ|. We have

\

D

|v|

2

e

−ψ

dV

n

≤ C

1

,

where C

1

is a constant (independent of ζ, f ). By a well known H¨ormander theorem, there exists a function u(z) which satisfies ∂u = v and

\

D

|u|

2

|z − ζ|

−2(n+1)

dV

n

≤ C

2

,

where C

2

is a constant (independent of ζ, f ). Then F (z) = η(z)f (z) − u(z) is a holomorphic function on D and kF k

D

≤ 1 + C

3

, where C

3

is also independent of ζ, f . Since η = 1 on V , u is holomorphic on V and we have

u(ζ) = 0, ∂u

∂z

j

(ζ) = 0, ∀1 ≤ j ≤ n.

So we have

F (ζ) = −u(ζ) = 0, ∂F

∂z

j

(ζ) = ∂f

∂z

j

(ζ), ∀1 ≤ j ≤ n.

Hence

ds

2D

(ζ; X) ≥ Cds

2D∩U

(ζ; X), ∀X and ζ ∈ D ∩ V, where C = (1 + C

3

)

−2

.

Theorem 1.1 is immediately derived from Lemmas 3.1–3.3 and the fol- lowing proposition:

Proposition 3.4 (cf. [9]). Let D be a bounded domain in C

n

and assume (1) the bounded holomorphic functions in D are dense in H

2

(D);

(2) lim

z→∂D

K

D

(z, z) = ∞.

Then the Bergman metric of D is complete.

Remark . In fact, we can replace assumption (1) by a weaker condition:

For each z

0

∈ ∂D and f ∈ H

2

(D), f can be approximated in L

2

-norm by a family of holomorphic functions on D which are bounded in some neigh- bourhood of z

0

.

This can be proved easily with the method of [14] (cf. p. 409).

Proof of Theorem 1.2. Let D be a bounded pseudoconvex domain with

Lipschitz boundary. For each z

0

∈ ∂D, there exist reals ε

0

, r

0

> 0 and

a vector T

0

that points outside of D such that z + εT

0

∈ C

n

\ D for all

z ∈ (C

n

\ D) ∩ B(z

0

, r

0

) and 0 < ε < ε

0

.

(7)

It is known from [5] that D admits a C

strictly psh function u on D such that u < 0 on D and lim

z→∂D

u(z) = 0. For ε, r > 0, we set

D

εr

= {z ∈ B(z

0

, r) | u(z − εT

0

) < 0}.

Put D

ε

= D

εr

0/2+ε

for ε < min{ε

0

, r

0

/4}. Then:

(1) D

ε

is pseudoconvex;

(2) D ∩ B(z

0

, r

0

/2) ⊂⊂ D

ε

;

(3) there exists a constant 0 < C < 1 such that Cε < d

Dε

(z) < ε

for all z ∈ ∂(D ∩ B(z

0

, r

0

/2)). Hence D locally satisfies condition S, and therefore Theorem 1.2 is a direct consequence of Theorem 1.1.

4. Bergman metric on hyperconvex domains. Let D be a bounded hyperconvex domain in C

n

, and u

D

(ζ, z) be the pluricomplex Green function on D. From [10] we know that there exists a C

strictly psh function u on D such that u < 0 on D and lim

z→∂D

u(z) = 0. We set D

t

= {z ∈ D | u(z) < t}. The following fact is due to Demailly:

Proposition 4.1 (cf. [5], p. 531). For each ζ ∈ D, u

D

(ζ, · ) is a contin- uous psh exhaustion function on D with values in [−∞, 0) and u

D

(ζ, z) ∼ log |z − ζ| as z → ζ.

Now we prove the following two lemmas:

Lemma 4.2. Suppose that D is a bounded hyperconvex domain in C

n

. Then for each ζ ∈ D and f ∈ H

2

(D), there exists a function F ∈ H

2

(D) satisfying F (ζ) = 0 and

kF − f k

D

≤ Ckf k

Dζ,−1

,

where D

ζ,t

= {z ∈ D | u

D

(ζ, z) < t} and C is a constant (independent of ζ, f ).

P r o o f. Fix ζ. There are negative C

strictly psh functions u

ε

defined on D

−a(ε)

, where a(ε) → 0 as ε → 0, such that u

ε

↓ u

D

(ζ, · ) as ε ↓ 0.

Let 0 ≤ κ ≤ 1 be a C

function on R such that κ(t) = 1 on (−∞, 1 − log 2) and κ(t) = 0 on [1, ∞). Put

ψ

ε

= − log(−u

ε

), ̺

ε

(z) = 1 − κ(ψ

ε

(z) + 1).

Again we have

|∂̺

ε

|

∂∂ψ

ε

≤ sup |κ

|.

(8)

Put g

ε

= ∂(̺

ε

f ) ∧ dz

1

∧ . . . ∧ dz

n

. One has kg

ε

k

2∂∂ψ

ε,2nuε

≤ 2

n

sup |κ

|

2

\

−2<uε<−1

|f |

2

e

−2nuε

dV

n

≤ 2

n

e

4n

sup |κ

|

2

kf k

2Dζ,−1

since u

ε

≥ u

D

(ζ, · ). By Proposition 2.1, there exists an (n, 0)-form h

ε

= eh

ε

dz

1

∧ . . . ∧ dz

n

on D

−a(ε)

such that ∂h

ε

= g

ε

and

ke h

ε

k

D

−a(ε),2nuε

≤ C

0

kf k

Dζ,−1

,

where C

0

is a constant (independent of ζ, ε, f ). Then F

ε

= ̺

ε

f − e h

ε

is holomorphic on D

−a(ε)

and

kF

ε

− f k

D−a(ε)

≤ k(1 − ̺

ε

)f k

D−a(ε)

+ ke h

ε

k

D−a(ε),2nuε

≤ (1 + C

0

)kf k

Dζ,−1

. We also have

kF

ε

k

D

−a(ε),2nuε

≤ k̺

ε

f k

D

−a(ε),2nuε

+ ke h

ε

k

D

−a(ε),2nuε

≤ (e

2n

+ C

0

)kf k

D

. Since u

ε

< 0, we can choose a sequence ε

j

→ 0 such that F

εj

converges weakly to a holomorphic function F on D. Hence

kF − f k

D

≤ (1 + C

0

)kf k

Dζ,−1

.

Since u

ε

decreases with ε, for each b > 0 and ε with 0 < a(ε) < b we have kF k

D−b,2nuε

≤ (C

0

+ e

2n

)kf k

D

.

Letting ε → 0 and then b → 0, one gets

kF k

D,2nuD(ζ, · )

≤ (C

0

+ e

2n

)kf k

D

. Since u

D

(ζ, z) ∼ log |z − ζ| as z → ζ, one has F (ζ) = 0.

Lemma 4.3. Suppose that D is a bounded hyperconvex domain in C

n

, and u

D

(ζ, z) is symmetric. Then the volume of D

ζ,−1

tends to zero as ζ → ∂D.

P r o o f. The proof is due to Ohsawa (cf. [12]). For any ε > 0, there exists a C

function κ

ε

(z) such that supp κ

ε

⊂ D

−ε/2

and κ

ε

|

D−ε

= 1. Then we can find a constant C = C(ε) such that Cu(z)+ κ

ε

(z) log |z − ζ| is a negative psh function on D for all ζ ∈ D

−ε

. Hence there exists a δ > 0 such that

u

D

(ζ, z) > −1, ∀ζ ∈ D

−ε

, z ∈ D \ D

−δ

.

Since u

D

(ζ, z) = u

D

(z, ζ), for any ε > 0 there exists a δ > 0 such that u

D

(ζ, z) > −1, ∀ζ ∈ D \ D

−δ

, z ∈ D

−ε

.

This proves the lemma.

To prove Theorem 1.3, we need the following proposition:

Proposition 4.4 (cf. [14], p. 408). Let D be a bounded domain in C

n

,

and {ζ

j

}

j=1

⊂ D a Cauchy sequence w.r.t. the Bergman metric. Then there

(9)

exist a subsequence {ζ

jk

}

k=1

and real numbers θ

k

such that

 K

D

( · , ζ

jk

) K

D1/2

jk

, ζ

jk

) e

k



∞ k=1

is a Cauchy sequence in H

2

(D).

Proof of Theorem 1.3. Suppose that D is not complete w.r.t. the Bergman metric. Then there exists a Cauchy sequence {ζ

j

}

j=1

w.r.t. the Bergman metric which converges to a boundary point ζ

0

of D as j → ∞. Proposition 4.4 gives us a subsequence {ζ

jk

}

k=1

and real numbers θ

k

satisfying

K

D

( · , ζ

jk

)

K

D1/2

jk

, ζ

jk

) e

k

→ f in H

2

(D) with kf k

D

= 1. This yields

|f (ζ

jk

)|

K

D1/2

jk

, ζ

jk

) = e

−iθk



f, K

D

( · , ζ

jk

) K

D1/2

jk

, ζ

jk

)



→ kf k

2

= 1.

By Lemma 4.2 there exists a sequence {F

k

}

k=1

of holomorphic functions on D satisfying F

k

jk

) = 0 and

kF

k

− f k

D

≤ Ckf k

Dζjk ,−1

k→∞

−→ 0 because Vol(D

ζjk,−1

) → 0 by Lemma 4.3. On the other hand

kF

k

− f k

D

≥ |f (ζ

jk

)|

K

D1/2

jk

, ζ

jk

) → 1, which is a contradiction.

5. A counterexample. Let ∆ be the unit disc in C and define D

N

= ∆ \

[

∞ k=1

∆(2

−k

, 2

−kN (k)

),

where ∆(x, r) is the disc centred at x with radius r > 0. Assume N (k) > 2 for all k. Then ∆(2

−k

, 2

−kN (k)

) are disjoint from each other for all k. We have the following criterion for the hyperconvexity of D

N

(cf. [12], p. 50):

Proposition 5.1. D

N

is hyperconvex if and only if P

k=1

N (k)

−1

= ∞.

We first prove the following:

Lemma 5.2. Let D be a bounded domain in C satisfying:

(∗) For each z

0

∈ ∂D, there exists a sequence {z

k

}

k=1

⊂ C \ D with z

k

→ z

0

as k → ∞ such that

|z

k

− z

0

|

2

log d(z

k

, ∂D) → 0 as k → ∞.

Then the Bergman metric on D is complete.

(10)

P r o o f. First, we prove K

D

(z, z) → ∞ as z → ∂D. For each z

0

∈ ∂D, put f

k

(z) = (z − z

k

)

−1

. Then f

k

is holomorphic on D and satisfies

kf

k

k

2D

\

D∩{z | |z−zk|>d(zk,∂D)}

|z − z

k

|

−2

dV

1

= 2π(C

1

− log d(z

k

, ∂D)), where C

1

is a constant depending only on the diameter of D.

For each z ∈ D ∩ ∆(z

0

, |z

k

− z

0

|), one has K

D

(z, z) ≥ |f

k

(z)|

2

kf

k

k

2D

≥ 1

8π|z

k

− z

0

|

2

(C

1

− log d(z

k

, ∂D)) . Thus we obtain K

D

(z, z) → ∞ as z → z

0

.

Next we want to show that every f ∈ H

2

(D) with kf k

D

≤ 1 can be approximated in L

2

-norm by a family {f

ε

}

ε>0

of holomorphic functions on D which are bounded in some open neighbourhood of z

0

.

For simplicity we can assume z

0

= 0 and D ⊂⊂ ∆. Put ψ(z) = − log(− log |z|).

Then ψ is a C

strictly psh function on ∆ \ {0} satisfying ∂∂ψ ≥ ∂ψ∂ψ.

For each 0 < ε < 1 we set

η

ε

= κ(−ψ − log log(1/ε) + 1), where κ is the function in the proof of Lemma 4.2. Put

D

ε

= {z ∈ C | 0 < |z| < ε/2} ∪ D.

Then η

ε

f ∈ C

(D

ε

). Since every domain in C is pseudoconvex, we can find a function y

ε

on D

ε

such that f

ε

= η

ε

f − y

ε

is a holomorphic function on D

ε

satisfying ky

ε

k

Dε

≤ C

2

kf k

D∩∆(0,ε1/2)

, where C

2

is a constant (independent of f, ε). This gives

kf

ε

− f k

D

≤ (1 + C

2

)kf k

D∩∆(0,ε1/2)

and kf k

Dε

≤ 1 + C

2

. Then f

ε

can be extended to a holomorphic function on D ∪ ∆(0, ε

1/2

) and kf

ε

− f k

D

→ 0 as ε → 0. The proof of Lemma 5.2 is complete by the remark after Proposition 3.4.

Lemma 5.2 implies

Lemma 5.3. Let D

N

be defined as above. Suppose that kN (k)2

−2k

→ 0 as k → ∞.

Then the Bergman metric on D

N

is complete.

Now we take N (k) = k

2

+1. Then D

N

is not hyperconvex by Proposition 5.1 and the Bergman metric on D

N

is complete by Lemma 5.3.

Acknowledgements. The author wishes to express his most sincere

gratitude and appreciation to Professor Jin-Hao Zhang.

(11)

References

[1] E. B e d f o r d and J. P. D e m a i l l y, Two counterexamples concerning the pluri- complex Green function in C

n

, Indiana Univ. Math. J. 37 (1988), 865–867.

[2] S. B e r g m a n, The Kernel Function and Conformal Mapping, 2nd ed., Amer. Math.

Soc., Providence, R.I., 1970.

[3] Z. B l o c k i, Smooth exhaustion functions in convex domains, Proc. Amer. Math.

Soc. 125 (1997), 477–484.

[4] H. J. B r e m e r m a n n, Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, in: Lectures on Functions of a Com- plex Variable, Univ. of Michigan Press, 1955, 349–383.

[5] J. P. D e m a i l l y, Mesures de Monge–Amp`ere et mesures pluriharmoniques, Math.

Z. 194 (1987), 519–564.

[6] K. D i e d e r i c h and T. O h s a w a, General continuity principles for the Bergman kernel , Internat. J. Math. 5 (1994), 189–199.

[7] H. G r a u e r t, Charakterisierung der Holomorphiegebiete durch die vollst¨ andige ahlersche Metrik, Math. Ann. 131 (1956), 38–75.

[8] L. H ¨ o r m a n d e r, An Introduction to Complex Analysis in Several Variables, North- Holland, 1990.

[9] M. J a r n i c k i and P. P f l u g, Bergman completeness of complete circular domains, Ann. Polon. Math. 50 (1989), 219–222.

[10] N. K e r z m a n et J.-P. R o s a y, Fonctions plurisousharmoniques d’exhaustion born´ees et domaines taut, Math. Ann. 257 (1981), 171–184.

[11] S. K o b a y a s h i, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290.

[12] T. O h s a w a, On the Bergman kernel of hyperconvex domains, Nagoya Math. J. 129 (1993), 43–52.

[13] —, On the completeness of the Bergman metric, Proc. Japan Acad. Ser. A Math.

Sci. 57 (1981), 238–240.

[14] P. P f l u g, Various applications of the existence of well growing holomorphic func- tions, in: Functional Analysis, Holomorphy and Approximation Theory, J. A. Bar- roso (ed.), North-Holland Math. Stud. 71, North-Holland, 1982, 391–412.

[15] W. Z w o n e k, On symmetry of the pluricomplex Green function for ellipsoids, Ann.

Polon. Math. 67 (1997), 121–129.

Institute of Mathematics Fudan University Shanghai 200433 P.R. China

E-mail: 96-165@fudan.edu.cn

Re¸ cu par la R´ edaction le 28.1.1998

evis´ e le 19.11.1998

Cytaty

Powiązane dokumenty

It is well known that any complete metric space is isomet- ric with a subset of a Banach space, and any hyperconvex space is a non- expansive retract of any space in which it

Hardy spaces consisting of adapted function sequences and generated by the q-variation and by the conditional q-variation are considered1. Their dual spaces are characterized and

Our purpose in this article is to give a brief and simple proof of this theorem in the general case of (not necessarily bounded) hyperbolic convex domains in C n.. In order for M to

Totally geodesic orientable real hypersurfaces M 2n+1 of a locally conformal Kaehler (l.c.K.) manifold M 2n+2 are shown to carry a naturally induced l.c.c.. manifolds in a natural

Within this approach, in [2]–[7], [13], [18], [21] qualified decay was obtained for the first boundary value problem for the wave equation in an exterior domain for which the

Following the spectacular result of Drury (“the union of two Sidon sets is a Sidon set”), a lot of improvements were achieved in the 70’s about such sets Λ.. Rider, in

Assume that all quasitilted algebras with less than n isomorphism classes of simple modules have a preprojective component, and let Λ be a quasitilted algebra with n ≥ 2

In this paper we obtain some equivalent characterizations of Bloch func- tions on general bounded strongly pseudoconvex domains with smooth boundary, which extends the known results