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# Completeness of the Bergman metric on non-smooth pseudoconvex domains

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(1)

n

n

D

2D

n j,k=1

2

D

j

k

j

k

n

2D

1

n

j

j

j

z∈∂D

Dj

Dj

j

j

j

αj

j

z∈∂D

Dj

[241]

(2)

0

0

0

0

0

n

n

D

ζ

n

D

n

n

(3)

2

2

0p,q

ds2

\

X

−ϕ

ds2

0p,q

ds2

2

ds2

1/2ds2

p,q

2

0p,q

ds2

∂∂ψ,ϕ

ϕ

∂∂ψ,ϕ

2ϕ

T

X

−ϕ

n

2

2

D

D,ϕ

2

−ϕ

2

2

D

j,t

j

Dj

j,t

Dj

j,t

j,Λj

j,t

j

(4)

## function satisfying χ|

(−∞,1+1/(2 log β))

[1,∞)

j

j

j

j

j

j,νj

Dj

j

j

j,νj

j

Dj

j

−1

−1/2

j

j

j

j,νj

j

j

j

j

j

j

j

j

j

j

j

j

∂∂ψ

j

j

∂∂ψ

j

j

j

j

j,Λj

j

j

j

1

n

j

j

−1/2

j

j

Dj

j

−1

j

j

j,µj

j

j

α

1/(2(1−α))

1/2j

(5)

j

2∂∂ψ

j,0

n

\

Dj

j

2∂∂ψ

j

2

n

n

2

2D\D

j,µj

k

j

j

1

n

j

j

j

j

Dj

D\Dj,µj

j

j

j

j

j

D

j

D

j

D

D\Dj,µj

D\Dδ

t

D

j

j

j,µj

j,δ/2

Dj

D

j(ε),µj(ε)

δ

j(ε)

D

z→∂D

D

z→∂D

D

n

0

0

2D

2D∩U

1,0

n

1,0

n

2D

D−1

2

2

D

(6)

n

n

2

D∩U

\

D

2

−ψ

n

1

1

\

D

2

−2(n+1)

n

2

2

D

3

3

j

j

j

2D

2D∩U

3

−2

n

2

z→∂D

D

0

2

2

0

0

0

0

0

0

n

n

0

0

0

(7)

z→∂D

εr

0

0

ε

εr

0/2+ε

0

0

ε

0

0

ε

Dε

0

0

n

D

z→∂D

t

D

D

n

2

2

D

Dζ,−1

ζ,t

D

ε

−a(ε)

ε

D

ε

ε

ε

ε

ε

∂∂ψ

ε

(8)

ε

ε

1

n

ε

2∂∂ψ

ε,2nuε

n

2

\

−2<uε<−1

2

−2nuε

n

n

4n

2

2Dζ,−1

ε

D

ε

ε

1

n

−a(ε)

ε

ε

ε

D

−a(ε),2nuε

0

Dζ,−1

0

ε

ε

ε

−a(ε)

ε

D−a(ε)

ε

D−a(ε)

ε

D−a(ε),2nuε

0

Dζ,−1

ε

D

−a(ε),2nuε

ε

D

−a(ε),2nuε

ε

D

−a(ε),2nuε

2n

0

D

ε

j

εj

D

0

Dζ,−1

ε

D−b,2nuε

0

2n

D

D,2nuD(ζ, · )

0

2n

D

D

n

D

ζ,−1

ε

ε

−ε/2

ε

D−ε

ε

−ε

D

−ε

−δ

D

D

D

−δ

−ε

n

j

j=1

(9)

jk

k=1

k

D

jk

D1/2

jk

jk

k

∞ k=1

2

j

j=1

0

jk

k=1

k

D

jk

D1/2

jk

jk

k

2

D

jk

D1/2

jk

jk

−iθk

D

jk

D1/2

jk

jk

2

k

k=1

k

jk

k

D

Dζjk ,−1

k→∞

ζjk,−1

k

D

jk

D1/2

jk

jk

N

∞ k=1

−k

−kN (k)

−k

−kN (k)

N

N

k=1

−1

0

k

k=1

k

0

k

0

2

k

(10)

D

0

k

k

−1

k

k

2D

## ≤

\

D∩{z | |z−zk|>d(zk,∂D)}

k

−2

1

1

k

1

0

k

0

D

k

2

k

2D

k

0

2

1

k

D

0

2

D

2

ε

ε>0

0

0

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

Dε

2

D∩∆(0,ε1/2)

2

ε

D

2

D∩∆(0,ε1/2)

Dε

2

ε

1/2

ε

D

N

−2k

N

2

N

N

## gratitude and appreciation to Professor Jin-Hao Zhang.

(11)

n

### R´evis´e le 19.11.1998

Cytaty

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