# Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type inC

Authors: Sanghyun Cho; Young Hwan You
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetΩ be a smoothly bounded pseudoconvex domain in C
3 and assume that z
0
∈
b
Ω is a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curve Γ
⊂
Ω
∪
{
z
0
}, connecting points  q
0
∈
Ω and z
0
∈
b
Ω, and a quantity M
(
z
,
X
), along z
∈
Γ, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.


---

## Body

## 1. Introduction

LetΩ be a smoothly bounded domain in C
n and let X be a holomorphic tangent vector at a point z in Ω, and let us denote the Bergman, Caratheodory, and Kobayashi metrics at z by B
Ω
(
z
;
X
), C
Ω
(
z
;
X
), and K
Ω
(
z
;
X
), respectively. When Ω is a strongly pseudoconvex domain in C
n, the optimal boundary behavior of the above metrics is well understood. For weakly pseudoconvex domains of finite type in C
n, several authors found some results about these metrics. But in each case, the lower bounds are different from the upper bounds [1–5]. In [6], Catlin got optimal estimates in a small constant and large constant sense for pseudoconvex domains of finite type in C
2. For pseudoconvex domains of finite type in C
n, the first author and Herbort extended Catlin’s result to the case that the Levi-form at z
0 has corank one [7, 8] or homogeneous finite diagonal type near z
0
∈
b
Ω [9, 10].To estimate the above invariant metrics, we need a complete geometric analysis nearz
0
∈
b
Ω of finite type, and then we construct a family of plurisubharmonic functions with maximal Hessian near b
Ω. However, this construction is really technical and known only for special types of domains mentioned above, but not for arbitrary pseudoconvex domains of finite type in C
n, even for n
=
3 case. Meanwhile, it is useful to understand the behavior of a holomorphic function near z
0
∈
b
Ω if we have precise estimates of the invariant metric along some curves.In the sequel, we letΩ be a smoothly bounded pseudoconvex domain in C
3 with smooth defining function r and let z
0
∈
b
Ω. Let M
(
z
0
)
=
(
1
,
m
,
m
3
) be Catlin’s multitype [11]. Thus, m
=
T
BG
(
z
0
) is the type in the sense of “Bloom-Graham.” If m
3
=
Δ
1
(
z
0
), then Ω is an h-extensible domain [12] and Herbort [10] got an estimate in this case. Here, Δ
q
(
z
0
) denotes finite q-type in the sense of D’Angelo. Thus, we assume that m
≤
m
3
<
Δ
1
(
z
0
). Regular finite 1-type at z
0
∈
b
Ω is the maximum order of vanishing of r
∘
γ for all one complex dimensional regular curves γ, γ
(
0
)
=
z
0 and γ
′
(
0
)
≠
0. We denote the regular finite 1-type at z
0 by T
Ω
reg
(
z
0
). Note that T
Ω
reg
(
z
0
) is a positive integer and T
Ω
reg
(
z
0
)
≤
Δ
1
(
z
0
).Assuming thatT
Ω
reg
(
z
0
)
=
η
<
∞, there exist coordinate functions z
=
(
z
1
,
z
2
,
z
3
) defined in a neighborhood V of z
0 such that z
0
=
0 and |
∂
r
/
∂
z
3
|
≥
c
0 on V for a uniform constant c
0
>
0, and |
r
(
z
1
,
0,0
)
| vanishes to order η, and (
∂
r
/
∂
z
2
)
(
0
)
=
0 (Theorem 2.1 in [13]). With these coordinates at hand, set
(1)
L
k
=
∂
∂
z
k
-
∂
r
∂
z
3
-
1
∂
r
∂
z
k
∂
∂
z
3
∶
=
∂
∂
z
k
+
e
k
(
z
)
∂
∂
z
3
,
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
k
=
1,2
,
L
3
=
∂
∂
z
3
.
Then, e
k
(
0
)
=
0, k
=
1,2, and {
L
1
,
L
2
,
L
3
} form a basis of C
T
(
1,0
)
(
V
) provided V is sufficiently small. For any integer j, k
>
0, set
(2)
L
j
,
k
∂
∂
¯
r
z
=
L
2
⋯
L
2
︸
j
-
1
times
L
¯
2
⋯
L
¯
2
︸
k
-
1
times
∂
∂
¯
r
z
L
2
,
L
¯
2
z
and define
(3)
C
l
z
=
max
⁡
L
j
,
k
∂
∂
¯
r
z
:
j
+
k
=
l
.
Let X
=
a
1
L
1
+
a
2
L
2
+
a
3
L
3 be a holomorphic tangent vector at z
∈
Ω and set
(4)
M
z
;
X
=
a
1
r
z
-
1
/
η
+
a
2
∑
l
=
2
m
‍
C
l
z
r
z
1
/
l
+
a
3
r
z
-
1
.
Let Γ
⊂
Ω
∪
{
z
0
} be the admissible curve defined in (20). Our main result is as follows.Theorem 1.
LetΩ
⊂
⊂
C
3 be a smoothly bounded pseudoconvex domain and assume z
0
∈
b
Ω is a point of finite 1-type in the sense of D’Angelo; that is, Δ
1
(
z
0
)
<
∞. Then, there exist a neighborhood V about z
0, an admissible curve Γ
⊂
Ω
∪
{
z
0
} connecting q
0
∈
Ω and z
0, and positive constants c and C such that, for all X
=
a
1
L
1
+
a
2
L
2
+
a
3
L
3 at z
∈
V
∩
Γ
∩
Ω,
(5)
c
M
z
;
X
≤
B
Ω
z
;
X
≤
C
M
z
;
X
c
M
z
;
X
≤
C
Ω
z
;
X
≤
C
M
z
;
X
c
M
z
;
X
≤
K
Ω
z
;
X
≤
C
M
z
;
X
.To prove Theorem1, we use special coordinates constructed in Section 2 of [13]. Thus, there is a special direction d, |
d
|
=
1, so that, for each δ
>
0, the two-dimensional slice D
δ
≔
{
(
z
2
,
z
3
)
;
r
(
d
δ
1
/
η
,
z
2
,
z
3
)
<
0
} becomes a pseudoconvex domain of finite type in C
2, whose type is less than or equal to m
=
T
BG
(
z
0
). We then apply the method which holds for the domains of finite type in C
2 as in [6]. To avoid the difficulty to push out the domain in z
1-direction, we use a bumping theorem of Cho [14].


## 2. Special Coordinates

LetΩ
⊂
C
3 and z
0
∈
b
Ω be as in Section 1. We may assume that z
0
=
0. In this section, we consider special coordinates defined near z
0
∈
b
Ω and then construct “balls" which are of maximal size on which r
(
z
) changes by no more than some prescribed number δ
>
0. In the following, we let α
=
(
α
1
,
α
2
) and β
=
(
β
1
,
β
2
) be multi-indices with respect to z
′
=
(
z
1
,
z
2
) variables. In Theorem 2.1 of [13], You constructed special coordinates which represent the local geometry of b
Ω near z
0.Theorem 2.
LetΩ be a smoothly bounded pseudoconvex domain in C
3 with smooth defining function r and assume T
Ω
r
e
g
(
0
)
=
η
<
∞, 0
∈
b
Ω. Then, there is a holomorphic coordinate system z
=
(
z
1
,
z
2
,
z
3
) about 0 such that
(6)
(
1
)
r
(
z
)
=
Re
z
3
+
∑
α
+
β
=
m
α
,
β
>
0
η
a
α
,
β
z
′
α
z
-
′
β
vvvvvvvvvvvvvvvvvvvv
+
O
z
3
z
+
z
′
η
+
1
,
2
r
t
,
0,0
≲
t
η
,
where z
′
=
(
z
1
,
z
2
) and where
(7)
a
α
,
β
≠
0
f
o
r
s
o
m
e
α
,
β
w
i
t
h
α
1
=
β
1
=
0
,
α
2
+
β
2
=
m
.Remark 3.
(1) The second condition in (6) and the property (7) say that r
(
z
) vanishes to order η along z
1 axis and order m along z
2 axis. These properties are crucial for the construction of maximal polydiscs Q
c
δ
(
z
δ
) contained in Ω.
(2) There are much more terms (mixed withz
1 and z
2 and their conjugates) in the summation part of (6) compared to the h-extensible domain cases.According to Proposition 2.6 and Remark 2.7 of [13], there are pairs of integers (
p
ν
,
q
ν
), ν
=
1
,
…
,
N, such that the terms satisfying α
1
+
β
1
=
p
ν and α
2
+
β
2
=
q
ν with α
2
>
0 and β
2
>
0 are dominant terms in the summation part of (6). Also, there is a small constant a
0
>
0 and a fixed direction d, |
d
|
=
1, in z
1 direction, such that, for each fixed δ
>
0 and for all z
1 satisfying |
z
1
-
d
δ
1
/
η
|
<
a
0
δ
1
/
η, those major terms in the summation part of (6) satisfy
(8)
∂
q
ν
∂
z
2
α
2
∂
¯
z
-
2
β
2
r
z
1
,
0,0
≈
z
1
p
ν
≈
δ
p
ν
/
η
,
where α
2
+
β
2
=
q
ν and where α
2
>
0 and β
2
>
0.Now, let us fixz
1 with |
z
1
-
d
δ
1
/
η
|
<
a
0
δ
1
/
η and consider the two-dimensional slice D
z
1
≔
{
(
z
2
,
z
3
)
:
r
(
z
1
,
z
2
,
z
3
)
<
0
}. For each z
=
(
z
1
,
0
,
z
3
) near b
Ω, set π
(
z
)
=
(
z
1
,
0
,
e
δ
)
≔
z
~
1
∈
b
Ω, where π
(
z
) is the projection of z onto b
Ω along z
3 direction. On D
z
1, following the argument in two-dimensional case as in the proof of Proposition 1.1 in [6], we construct special coordinates ζ
=
(
ζ
1
,
ζ
2
,
ζ
3
)
=
(
z
1
,
z
2
,
ζ
3
) about z
~
1 so that, in terms of new coordinates, there are no pure terms in z
2 variable in the expression of r
(
z
) in (6).Proposition 4.
For each fixedz
~
1
=
(
z
1
,
0
,
e
δ
)
∈
V
∩
b
Ω, there exists a holomorphic coordinate system z
=
Φ
z
~
1
(
ζ
)
=
(
z
1
,
z
2
,
Φ
3
(
ζ
)
), ζ
=
(
ζ
1
,
ζ
2
,
ζ
3
)
=
(
z
1
,
z
2
,
ζ
3
), where Φ
3
(
ζ
) is defined by
(9)
Φ
3
ζ
=
e
δ
+
∂
r
∂
z
3
z
~
1
-
1
×
ζ
3
2
-
∑
l
=
2
m
‍
c
l
z
~
1
ζ
2
l
-
∂
r
∂
z
2
z
~
1
ζ
2
≔
e
δ
+
d
0
z
~
1
ζ
3
+
∑
l
=
1
m
‍
d
l
(
z
~
1
)
ζ
2
l
,
and the function ρ, given by ρ
(
z
1
,
ζ
′′
)
≔
r
∘
Φ
z
~
1
(
z
1
,
ζ
′′
), ζ
′′
=
(
ζ
2
,
ζ
3
), satisfies
(10)
ρ
z
1
,
ζ
′′
=
Re
Φ
3
ζ
+
∑
j
+
k
=
2
j
,
k
>
0
m
a
j
,
k
z
~
1
ζ
2
j
ζ
¯
2
k
+
E
ζ
,
where
(11)
E
ζ
=
O
Φ
3
ζ
ζ
+
∑
ν
=
1
N
‍
z
1
1
+
p
ν
ζ
2
q
ν
+
ζ
2
m
+
1
.In view of (6) and (8), the major terms in (10) are a
j
,
k
(
z
~
1
)
ζ
2
j
ζ
¯
2
k where j
+
k
=
α
2
+
β
2
=
q
ν for some α
2 and β
2 with α
2
>
0 and β
2
>
0. Also, from (8), it follows that
(12)
a
j
,
k
z
~
1
ζ
2
j
ζ
¯
2
k
≈
z
1
p
ν
z
2
q
ν
,
and these terms control the error terms |
z
1
|
1
+
p
ν
|
ζ
2
|
q
ν in E
(
ζ
). As in Section 1 in [6], set
(13)
A
l
z
~
1
=
max
⁡
a
j
,
k
z
~
1
;
j
+
k
=
l
,
l
=
2
,
…
,
m
,
and for each sufficiently small δ
>
0, we set
(14)
τ
z
~
1
,
δ
=
min
⁡
δ
A
l
z
~
1
1
/
l
;
2
≤
l
≤
m
.
Thus, for all z
1 with |
z
1
-
d
δ
1
/
η
|
<
a
0
δ
1
/
η, it follows from (8) and (14) that
(15)
τ
z
~
1
,
δ
≲
δ
z
1
p
ν
1
/
q
ν
,
ν
=
1
,
…
,
N
,
and hence the summation part of (10) is dominated by C
δ.For eachz
~
=
(
z
1
,
0
,
z
3
) near b
Ω, set ζ
~
=
Φ
z
~
1
-
1
(
z
~
)
=
(
z
1
,
0
,
ζ
~
3
), where Φ
z
~
1 is the function defined in Proposition 4. For each small e
>
0, set
(16)
R
e
δ
ζ
~
=
ζ
3
-
ζ
~
3
ζ
:
ζ
1
-
z
1
<
e
δ
1
/
η
,
ζ
2
<
e
τ
z
~
1
,
δ
,
ζ
3
-
ζ
~
3
<
e
δ
,
Q
e
δ
(
z
~
)
=
z
:
z
=
Φ
z
~
1
ζ
,
ζ
∈
R
e
δ
ζ
~
.
For each σ
>
0, let Ω
σ
=
{
z
;
r
(
z
)
<
σ
} and define
(17)
S
σ
=
z
∈
V
:
-
σ
<
r
z
≤
σ
S
-
(
σ
)
=
z
∈
V
:
-
σ
<
r
z
≤
0
,
and set z
~
δ
=
(
d
δ
1
/
η
,
0
,
e
δ
)
∈
b
Ω, where z
1 is replaced by d
δ
1
/
η in z
~
1
=
(
z
1
,
0
,
e
δ
). The following theorem is about the existence of plurisubharmonic function with maximal Hessian. In [6], for the domains in C
2, Catlin constructed the functions with maximal Hessian on the strip S
(
δ
)
∩
V. However, for regular finite type pseudoconvex domains in C
3, we show that the functions have maximal Hessian on each ball Q
b
δ
(
z
~
δ
) and this will suffice to prove the boundary behavior of the invariant metrics. The proof of the following theorem can be found in Theorem 3.2 in [9].Theorem 5.
There is a small constantb
>
0 such that, for each small δ
>
0, there is a plurisubharmonic function g
δ
∈
C
0
∞
(
Q
2
b
δ
(
z
~
δ
)
) with the following properties:(i)
|
g
δ
(
ζ
)
|
≤
1, z
∈
Ω
δ,
(ii)
for allL
=
b
1
L
1
+
b
2
L
2
+
b
3
L
3 at z, where z
∈
Q
b
δ
(
z
~
δ
)
∩
S
(
b
δ
),
(18)
∂
∂
¯
g
δ
L
,
L
¯
z
≳
δ
-
2
/
η
b
1
2
+
τ
z
~
δ
,
δ
-
2
b
2
2
+
δ
-
2
b
3
2
,
(iii)
ifΦ
(
ζ
)
=
(
ζ
1
,
ζ
2
,
Φ
3
(
ζ
)
), where Φ
3 is defined in (10) for z
~
δ, then
(19)
D
~
α
g
δ
∘
Φ
ζ
≤
C
α
δ
-
α
1
/
η
τ
z
~
δ
,
δ
-
α
2
δ
-
α
3 holds for all ζ
∈
R
2
b
δ
(
z
~
δ
), where D
~
α
=
D
~
1
α
1
D
~
2
α
2
D
~
3
α
3.LetΓ
⊂
Ω be a curve defined by
(20)
Γ
≔
z
δ
:
z
δ
=
d
δ
1
/
η
,
0
,
e
δ
-
b
δ
2
,
0
≤
δ
≤
δ
0
,
for sufficiently small δ
0
>
0 and b
>
0. In the sequel, for each z
δ
=
(
d
δ
1
/
η
,
0
,
e
δ
-
b
δ
/
2
)
∈
Γ, set ζ
δ
≔
Φ
z
~
δ
-
1
(
z
δ
) and set Ω
~
=
Φ
z
~
δ
-
1
(
Ω
). In view of Proposition 3.4 in [9], there is a uniform small constant c
>
0 such that R
c
δ
(
ζ
δ
)
⊂
⊂
R
b
δ
(
z
~
δ
)
∩
Ω
~, and hence
(21)
Q
c
δ
z
δ
=
z
:
z
=
Φ
z
~
δ
ζ
,
ζ
∈
R
c
δ
ζ
δ
⊂
⊂
Q
b
δ
z
~
δ
∩
Ω
,
provided c
>
0 and δ
0
>
0 are sufficiently small. In particular, we have Γ
⊂
Ω
∪
{
z
0
}. Note that τ
(
z
δ
,
δ
)
≈
τ
(
z
~
δ
,
δ
), and for z
∈
Q
c
δ
(
z
δ
)
⊂
Ω, we note that |
r
(
z
)
|
≈
δ. Thus, as in Proposition 1.3 and Corollary 1.4 in [6], we obtain that
(22)
τ
z
δ
,
δ
-
1
≈
∑
k
=
2
m
‍
C
k
z
r
z
1
/
k
,
z
∈
Q
c
δ
z
δ
,
where C
k
(
z
) is defined in (3). In the sequel, we set τ
1
=
δ
1
/
η, τ
2
=
τ
(
z
~
δ
,
δ
), and τ
3
=
δ. If we use the plurisubharmonic weight functions constructed in Theorem 5 and follow the method to prove Theorem 6.1 in [6], we get the following estimates of the Bergman kernel along Γ.Theorem 6.
Letz
0
∈
b
Ω be a point of regular finite 1-type and T
Ω
r
e
g
(
z
0
)
=
η. Then, K
Ω
(
z
δ
,
z
δ
), the Bergman kernel function of Ω at z
δ
∈
Γ, δ
>
0, satisfies
(23)
K
Ω
z
δ
,
z
δ
≈
δ
-
2
τ
1
-
2
τ
2
-
2
.


## 3. Metric Estimates

In this section, we estimate the behavior of the invariant metric alongΓ. In [15], Hahn got the following inequalities:
(24)
C
Ω
z
;
X
≤
B
Ω
z
;
X
,
C
Ω
z
;
X
≤
K
Ω
z
;
X
.
Therefore, the estimates for the lower bounds of C
Ω
(
z
;
X
) will suffice for the lower bounds of B
Ω
(
z
;
X
) and K
Ω
(
z
;
X
). First, we recall the following bumping theorem [14].Theorem 7 (Theorem 2.3 in [14]).
Letz
0 be a point of finite 1-type in the boundary of a pseudoconvex domain Ω
⊂
C
n, defined by Ω
=
{
z
:
r
(
z
)
<
0
}. Then, there exist V
∋
z
0 and a smooth 1-parameter family of pseudoconvex domains Ω
t, 0
≤
t
<
t
0, each defined by Ω
t
=
{
z
;
r
(
z
,
t
)
<
0
}, where r
(
z
,
t
) has the following properties:(1)
r
(
z
,
t
) is smooth in z for z near b
Ω and in t for 0
≤
t
<
t
0;
(2)
r
(
z
,
t
)
=
r
(
z
), for z
∉
V;
(3)
∂
r
/
∂
t
(
z
,
t
)
≤
0;
(4)
r
(
z
,
0
)
=
r
(
z
);
(5)
forz in V, ∂
r
/
∂
t
<
0.Proof of Theorem1.
In the sequel, let us fixδ
>
0 and, for each z
δ
∈
Γ, set π
(
z
δ
)
=
z
~
δ
=
(
d
δ
1
/
η
,
0
,
e
δ
)
∈
b
Ω and consider the special coordinates ζ
=
(
z
1
,
z
2
,
ζ
3
) and Φ
z
~
δ
(
ζ
)
=
(
z
1
,
z
2
,
Φ
3
(
ζ
)
)
=
z, where Φ
3 is defined in Proposition 4. From (9), we see that ζ
δ
=
(
d
δ
1
/
η
,
0
,
-
b
δ
/
2
d
0
(
z
~
δ
)
)
≔
(
ζ
~
1
,
ζ
~
2
,
ζ
~
3
). We will estimate the metrics at ζ
δ. For all small δ
>
0 and for each ζ
′′
=
(
ζ
2
,
ζ
3
), define
(25)
J
δ
ζ
′′
=
δ
2
+
ζ
3
2
+
∑
k
=
2
m
‍
A
k
z
~
δ
2
ζ
2
2
k
1
/
2
,
where A
k
(
z
~
δ
) is defined in (13) with z
~
1 replaced by z
~
δ. Let c
>
0 be the fixed constant determined in (21). Note that Φ
z
~
δ
(
d
δ
1
/
η
,
0,0
)
=
z
~
δ. Set
(26)
Ω
~
a
,
δ
=
ζ
;
ζ
1
-
d
δ
1
/
η
<
c
δ
1
/
η
,
ζ
2
<
a
,
ζ
3
<
a
,
ρ
ζ
1
,
ζ
2
,
ζ
3
<
0
ζ
ζ
1
-
d
δ
1
/
η
<
c
δ
1
/
η
,
ζ
2
<
a
,
ζ
3
<
a
,
,
and, for each ϵ
>
0, define
(27)
Ω
~
a
,
δ
ϵ
=
ζ
;
ζ
1
-
d
δ
1
/
η
<
c
δ
1
/
η
,
ζ
2
<
a
,
ζ
3
<
a
,
ρ
d
δ
1
/
η
,
ζ
′′
<
ϵ
J
δ
ζ
′′
,
and for all small e
>
0 set B
e
=
R
e
δ
(
ζ
δ
). By (21), we see that ζ
δ
∈
B
e
⊂
Ω
~ for all e
≤
c. Note that the domains Ω
~
a
,
δ
ϵ are pushed out only in ζ
2 and ζ
3 directions but not in ζ
1 direction. To avoid the difficulty to push out Ω
~ in ζ
1 direction, we use a bumping family of Theorem 7. Consider a bumping family of pseudoconvex domains {
Ω
t
}
0
≤
t
≤
t
0 with front V and set D
=
Ω
t
0. For each r
>
0, let U
r
(
z
) be a ball of radius r
>
0 with center at z and set U
~
r
(
ζ
)
=
Φ
z
~
δ
-
1
(
U
r
(
Φ
z
~
δ
(
ζ
)
)
). Then, there is r
0
>
0 such that
(28)
Q
c
δ
z
δ
⊂
Ω
a
,
δ
ϵ
=
Φ
z
~
δ
Ω
~
a
,
δ
ϵ
⊂
U
r
0
/
4
(
0
)
⊂
U
r
0
(
0
)
⊂
⊂
D
,
for all sufficiently small a
>
0, ϵ
>
0, and δ
>
0.
In view of the proof in Section 3 of [13], we have Ω
~
a
,
δ
⊂
Ω
~
a
,
δ
ϵ
/
2
⊂
Ω
~
a
,
δ
ϵ and there is a uniformly (independent of δ
>
0) bounded function f
~
=
f
~
(
ζ
2
,
ζ
3
) which is holomorphic on Ω
~
a
,
δ
ϵ and satisfies
(29)
Y
′′
f
~
ζ
δ
≳
b
2
τ
2
-
1
+
b
3
τ
3
-
1
,
where Y
′′
=
b
2
(
∂
/
∂
ζ
2
)
+
b
3
∂
/
∂
ζ
3. Here, we may assume that f
~
(
0
,
-
b
δ
/
d
0
(
z
~
δ
)
)
=
0. In the sequel, we let Y be a vector field given by Y
=
b
1
(
∂
/
∂
ζ
1
)
+
b
2
(
∂
/
∂
ζ
2
)
+
b
3
(
∂
/
∂
ζ
3
). If |
b
1
|
τ
1
-
1
≥
|
b
2
|
τ
2
-
1
+
|
b
3
|
τ
3
-
1, then set v
δ
=
τ
1
-
1
(
ζ
1
-
d
δ
1
/
η
). Otherwise, set v
δ
=
f
~
(
ζ
2
,
ζ
3
). From (29), we note that
(30)
Y
v
δ
ζ
δ
≳
∑
i
=
1
3
‍
b
i
τ
i
-
1
.
Letψ
∈
C
0
∞
(
U
), where U is the unit polydisc in C
3, such that ψ
(
z
)
=
1 if |
z
i
|
≤
1
/
2, i
=
1,2
,
3, and set
(31)
ψ
d
ζ
=
ψ
ζ
1
-
ζ
~
1
d
τ
1
,
ζ
2
d
τ
2
,
ζ
3
-
ζ
~
3
d
τ
3
,
and set β
δ
=
v
δ
ψ
d. Then, β
δ
(
ζ
δ
)
=
0. Since f
~ is bounded independent of δ (and hence independent of ζ
δ), there exists a constant C
>
0, independent of δ, such that |
β
δ
|
≤
C. We want to correct β
δ so that the corrected function f
δ becomes a uniformly bounded holomorphic function on Ω
~ satisfying the estimate (30) with β
δ replaced by f
δ. With bumped domain D
=
Ω
t
0 at hand, set D
~
=
Φ
z
~
δ
-
1
(
D
). On D
~, instead of Ω
~, we will employ weighted estimates of ∂
¯ that is essentially a replication of the proof of Theorem 6.1 in [6].
Letg
δ be the weight function defined in Theorem 5 and set g
~
δ
=
Φ
z
~
δ
*
g
δ. By replacing g
~
δ by g
~
δ
+
|
ζ
|
2
≔
ϕ, we can obviously assume that ϕ is strictly plurisubharmonic on D
~ and ϕ
(
ζ
δ
)
=
0. In view of Theorem 5, we also have
(32)
∂
∂
¯
ϕ
Y
,
Y
¯
ζ
≳
τ
1
-
2
b
1
2
+
τ
2
-
2
b
2
2
+
τ
3
-
2
b
3
2
,
ζ
∈
R
c
δ
ζ
δ
.
From property (iii) in Theorem 5, there is a small constant a, 0
<
a
≤
c (independent of τ
i, i
=
1,2
,
3), so that
(33)
ϕ
(
ζ
)
≥
2
Re
h
ζ
+
a
∑
i
=
1
3
‍
τ
i
-
2
ζ
i
-
ζ
~
i
2
,
ζ
∈
R
c
δ
ζ
δ
,
where
(34)
h
ζ
=
∑
i
=
1
3
‍
∂
ϕ
∂
ζ
i
ζ
δ
ζ
i
-
ζ
~
i
+
1
2
∑
i
,
j
=
1
3
‍
∂
2
ϕ
∂
ζ
i
∂
ζ
j
ζ
δ
ζ
i
-
ζ
~
i
ζ
j
-
ζ
~
j
.
If we set a
~
=
a
3
/
3, it follows, from (33), that
(35)
Re
h
ζ
≤
-
a
~
,
ζ
∈
ζ
;
ϕ
z
≤
a
~
∩
supp
⁡
∂
¯
ψ
d
.
In the sequel, we setB
e
=
R
e
δ
(
ζ
δ
) for each small e
>
0. For each s
≥
0, set
(36)
α
s
=
∂
¯
β
δ
e
s
h
=
v
δ
e
s
h
∂
¯
ψ
d
ζ
≔
∑
i
=
1
3
‍
α
s
,
i
d
ζ
i
¯
.
Then, α
s is a ∂
¯-closed smooth (
0,1
)-form with supp
⁡
α
s
⊂
R
c
δ
(
ζ
δ
)
=
B
c. Let χ be a smooth convex increasing function that satisfies χ
(
t
)
=
0 for t
≤
a
~
/
2 and χ
′′
(
t
)
>
0 for t
>
a
~
/
2. Now, define
(37)
λ
s
ζ
=
ϕ
ζ
+
s
2
χ
ϕ
ζ
.
According to the weighted estimates of ∂
¯-equation on D
~ (instead of Ω
~) and by using estimate (32) for each s
≥
0, there is u
s which satisfies ∂
¯
u
s
=
α
s, and
(38)
u
s
λ
s
≲
∫
D
~
-
B
c
‍
α
s
2
e
-
λ
s
+
∫
B
c
‍
∑
i
=
1
3
‍
τ
i
2
α
s
,
i
2
e
-
λ
s
d
V
.
Since |
α
s
,
i
|
≲
e
s
Re
h
τ
i
-
1 and supp
⁡
α
s
⊂
B
c, it follows from (38) that
(39)
∫
D
~
‍
u
s
2
e
-
λ
s
d
V
≲
∫
B
c
‍
∑
i
=
1
3
‍
τ
i
2
α
s
,
i
2
e
-
λ
s
d
V
≲
∫
supp
⁡
∂
¯
ψ
d
‍
e
2
s
Re
h
-
ϕ
-
s
2
χ
ϕ
d
V
.
We consider the integrand of the last integral. If ϕ
(
z
)
≥
a
~, then χ
(
ϕ
(
z
)
)
≥
χ
(
a
~
)
>
0, so the s
2-term in the exponent predominates. On the other hand, if z
∈
supp
⁡
∂
¯
ψ
d and ϕ
(
z
)
≤
a
~, then (35) shows that the integrand tends to zero. Thus, for any ϵ
0
>
0, there exist s
0
>
0 and a function u
s
0 so that ∂
¯
u
s
0
=
α
s
0 and
(40)
∫
D
~
‍
u
s
0
2
e
-
λ
s
0
d
V
≲
∫
supp
⁡
∂
¯
ψ
d
‍
ϵ
0
d
V
≲
ϵ
0
∏
i
=
1
3
‍
τ
i
2
.
Sinceϕ
(
ζ
δ
)
=
0, it follows, from the property (iii) of Theorem 5, that there is e
>
0, independent of ζ
δ, such that ψ
d
(
z
)
=
1 and ϕ
(
z
)
<
a
~
/
2 for all z
∈
B
e. Note that λ
s is independent of s for z
∈
B
e, and u
s
0 is holomorphic in B
e. By mean value theorem, we have
(41)
∂
u
s
0
∂
ζ
k
ζ
δ
2
≲
τ
k
-
2
∏
i
=
1
3
‍
τ
i
-
2
∫
B
e
‍
u
s
0
2
e
-
λ
s
0
d
V
≲
ϵ
0
τ
k
-
2
,
k
=
1,2
,
3
,
and hence it follows that
(42)
Y
u
s
0
ζ
δ
≲
ϵ
0
max
⁡
b
k
τ
k
-
1
.
Now, set f
δ
=
β
δ
e
s
0
h
-
u
s
0. Then, f
δ is holomorphic on D
~
=
Φ
z
~
δ
-
1
(
D
). Since β
δ
(
ζ
δ
)
=
h
(
ζ
δ
)
=
0, it follows, from (30) and (42), that f
δ satisfies
(43)
Y
f
δ
ζ
δ
≳
∑
i
=
1
3
‍
b
i
τ
i
-
1
,
provided ϵ
0 is sufficiently small.
We want to show thatsu
p
Ω
~
|
f
δ
|
≤
C, where C
>
0 is independent of δ. Recall that s
0
>
0 is fixed. Thus, from the property (iii) of Theorem 5, there is a uniform constant C
0
>
0 such that |
β
δ
e
s
0
h
|
≤
C
0. Let r
0
>
0 be the constant satisfying (28) and assume that ζ
∈
U
~
r
0
/
2
(
0
)
=
Φ
z
~
δ
-
1
(
U
r
0
/
2
(
0
)
). Since f
δ is holomorphic on D
~, it follows, by (40) and mean value theorem, that there exists a constant C
1
>
0, independent of δ
>
0, such that
(44)
f
δ
ζ
2
≲
r
0
-
6
∫
U
~
r
0
/
2
(
ζ
)
‍
f
δ
2
d
V
≤
C
1
.
We need to show the boundedness of f
δ outside U
~
r
0
/
2
(
0
). Let χ
1 and χ
2 be smooth cutoff functions with
(45)
i
χ
1
(
z
)
=
1
if
z
≥
r
0
2
,
χ
2
z
=
1
if
z
∈
supp
⁡
χ
1
ii
χ
2
z
=
0
if
z
≤
r
0
4
,
and set χ
~
i
=
Φ
z
~
δ
*
(
χ
i
), i
=
1,2. By Kohn’s theorem on global regularity for the ∂
¯-equation, the following estimate for the solution of ∂
¯
u
=
α,
(46)
χ
~
1
u
s
0
4
2
≲
χ
~
2
α
s
0
4
2
+
u
s
0
2
,
holds on D provided s
0
>
0 is sufficiently large. Note that χ
~
2
α
s
0
=
0 because supp
⁡
α
s
0
⊂
R
c
δ
(
ζ
δ
)
⊂
U
~
r
0
/
4
(
0
) for all sufficiently small δ
>
0. Thus, we conclude from (40), (46), and the Sobolev lemma that
(47)
su
p
D
~
χ
~
1
u
s
0
≲
χ
~
1
u
s
0
4
2
≲
u
s
0
2
≤
C
2
,
where C
2 is independent of δ.
Combining (44) and (47) and by the fact that |
β
δ
e
s
0
h
|
≤
C
0, we conclude that
(48)
su
p
D
~
f
δ
≤
C
,
where C is independent of ζ
δ and δ. Therefore, it follows from (43) and (48) that
(49)
C
Ω
~
ζ
δ
;
Y
≥
C
D
~
ζ
δ
;
Y
≳
∑
i
=
1
3
‍
b
i
τ
i
-
1
.
On the other hand, the polydisc B
c
=
R
c
δ
(
ζ
δ
) about ζ
δ lies in Ω
~. So one obtains that
(50)
C
Ω
~
ζ
δ
;
Y
≤
C
B
c
ζ
δ
;
Y
=
max
⁡
b
k
c
τ
k
-
1
:
k
=
1,2
,
3
.
Thus, one concludes from (49) and (50) that
(51)
C
Ω
~
ζ
δ
;
Y
≈
∑
i
=
1
3
‍
b
i
τ
i
-
1
.
SetL
k
′
=
(
d
Φ
z
~
δ
-
1
)
L
k, k
=
1,2
,
3, where L
k’s are defined in (1) in terms of z-coordinates defined in Theorem 1. At ζ
δ
=
(
d
δ
1
/
η
,
0
,
-
b
δ
/
d
0
(
z
~
δ
)
), from the holomorphic coordinate change of Φ
z
~
δ in Proposition 4, we see that
(52)
L
1
′
=
∂
∂
ζ
1
+
e
1
z
δ
d
0
z
~
δ
∂
∂
ζ
3
≔
∂
∂
ζ
1
+
e
~
1
z
δ
∂
∂
ζ
3
,
L
2
′
=
∂
∂
ζ
2
+
d
1
z
~
δ
+
e
2
z
δ
∂
∂
ζ
3
≔
∂
∂
ζ
2
+
e
~
1
z
δ
∂
∂
ζ
3
,
and
that
L
3
′
=
d
0
z
~
δ
∂
∂
ζ
3
,
where d
0
(
z
~
δ
)
=
(
1
/
2
)
(
(
∂
r
/
∂
z
3
)
(
z
~
δ
)
)
-
1 and d
1
(
z
~
δ
)
=
-
(
(
∂
r
/
∂
z
3
)
(
z
~
δ
)
)
-
1
(
∂
r
/
∂
z
2
)
(
z
~
δ
) and where e
i
=
-
(
∂
r
/
∂
z
3
)
-
1
(
∂
r
/
∂
z
i
), i
=
1,2. Since (
∂
r
/
∂
z
i
)
(
0
)
=
0, i
=
1,2, and |
∂
r
/
∂
z
3
|
≈
1, independent of δ, it follows that |
e
~
i
|
≲
δ, i
=
1,2. Thus, if the vector Y
=
∑
i
=
1
3
‍
b
i
(
∂
/
∂
ζ
i
) is written as Y
=
∑
i
=
1
3
‍
a
i
L
i
′, then it follows that
(53)
max
⁡
b
i
τ
i
-
1
≈
∑
i
=
1
3
‍
a
i
τ
i
-
1
.
Let us writeX
=
∑
i
=
1
3
‍
a
i
L
i, and Y
=
(
Φ
z
~
δ
-
1
)
*
X
=
∑
i
=
1
3
‍
a
i
L
i
′
=
∑
i
=
1
3
‍
b
i
(
∂
/
∂
ζ
i
). From (51), (53), and the invariance property of the metric, it follows that
(54)
C
Ω
z
δ
;
X
=
C
Ω
~
ζ
δ
;
Y
≈
∑
i
=
1
3
‍
a
i
τ
i
-
1
.
To obtain an upper bound for the Bergman metric, we note thatR
c
δ
(
ζ
δ
)
⊂
Ω
~. Thus, by elementary estimates, for any f
∈
A
2
(
Ω
~
)
≔
L
2
(
Ω
~
)
∩
A
(
Ω
~
), we obtain that
(55)
∂
f
∂
ζ
k
ζ
δ
2
≲
τ
k
-
2
∏
j
=
1
3
‍
τ
j
-
2
f
L
2
(
Ω
~
)
2
,
for k
=
1,2
,
3. Therefore, it follows that
(56)
b
Ω
~
ζ
δ
;
Y
≲
∑
k
=
1
3
‍
b
k
τ
k
-
1
∏
j
=
1
3
‍
τ
j
-
1
,
where
(57)
b
Ω
~
ζ
δ
;
Y
=
sup
⁡
Y
f
ζ
δ
:
f
∈
A
2
Ω
~
,
f
z
=
0
,
f
L
2
Ω
~
≤
1
.
Combining (23) and (56), one concludes that
(58)
B
Ω
~
ζ
δ
;
Y
=
b
Ω
~
ζ
δ
;
Y
K
Ω
~
ζ
δ
,
ζ
δ
1
/
2
≲
∑
k
=
1
3
‍
b
k
τ
k
-
1
.
To estimate the upper bound of the Kobayashi metric, set(59)
R
=
min
⁡
c
τ
k
b
k
-
1
:
k
=
1,2
,
3
.
Then,
(60)
f
t
=
b
1
t
,
b
2
t
,
-
b
δ
2
+
b
3
t
defines a map f
:
D
R
⊂
C
→
B
c
=
R
c
δ
(
ζ
δ
)
⊂
Ω
~ with f
*
(
(
∂
/
∂
t
)
0
)
=
Y
=
∑
k
=
1
3
‍
b
k
(
∂
/
∂
ζ
k
). Hence,
(61)
K
Ω
~
ζ
δ
;
Y
≤
K
B
c
ζ
δ
;
Y
≤
R
-
1
≤
max
⁡
⁡
b
k
c
τ
k
-
1
:
k
=
1,2
,
3
≲
∑
k
=
1
3
‍
b
k
τ
k
-
1
.
Combining (51), (58), and (61), we obtain that
(62)
C
Ω
~
ζ
δ
;
Y
≈
B
Ω
~
ζ
δ
;
Y
≈
K
Ω
~
ζ
δ
;
Y
≈
∑
i
=
1
3
‍
b
i
τ
i
-
1
,
and hence the invariance property implies that
(63)
C
Ω
z
δ
;
X
≈
B
Ω
z
δ
;
X
≈
K
Ω
z
δ
;
X
≈
∑
i
=
1
3
‍
a
i
τ
i
-
1
.
If we combine (3), (4), (22), and (63), a proof of Theorem 1 is completed.



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*Source: 697160-2014-11-12.xml*