# Some Topics Related to Bergman Kernel

YIN Weiping  YIN Weiping: Dept. of Math., Capital Normal Univ., Beijing
100037, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
Actually we will discuss some topics related to Bergman kernel on Cartan-
Hartogs domain. Cartan-Hartogs domain is introduced by Guy Roos and Weiping
YIN in 1998, which is built on the Cartan domains(classical domains). The four
big types of Cartan domains can be written as[1]:

$\begin{array}[]{ll}R_{I}(m,n)&:=\\{Z\in{\bf{C}^{mn}}:I-Z\overline{Z}^{t}>0,\\},\\\\[8.53581pt]
R_{II}(p)&:=\\{Z\in{\bf{C}^{p(p+1)/2}}:I-Z\overline{Z}^{t}>0,\\},\\\\[8.53581pt]
R_{III}(q)&:=\\{Z\in{\bf{C}^{q(q-1)/2}}:I-Z\overline{Z}^{t}>0,\\},\\\\[8.53581pt]
R_{IV}(n)&:=\\{Z\in{\bf{C}^{n}}:1+|ZZ^{t}|^{2}-2Z\overline{Z}^{t}>0,\\\
&1-|ZZ^{t}|^{2}>0\\}.\end{array}$

where $Z$ is $m\times n$ matrix, $p$ degree symmetric matrix, $q$ degree skew
symmetric matrix and $n$ dimensional complex vector respectively. Then the
Cartan-Hartogs domains can be introduced as follows:

$Y_{I}:=Y_{I}(N,m,n;K):=\\{W\in{\bf{C}^{N}},Z\in R_{I}(m,n):$
$|W|^{2K}<\det(I-Z\overline{Z}^{t}),K>0\\},$
$Y_{II}:=Y_{II}(N,p;K):=\\{W\in{\bf{C}^{N}},Z\in R_{II}(p):$
$|W|^{2K}<\det(I-Z\overline{Z}^{t}),K>0\\},$
$Y_{III}:=Y_{III}(N,q;K):=\\{W\in{\bf{C}^{N}},Z\in R_{III}(q):$
$|W|^{2K}<\det(I-Z\overline{Z}^{t}),K>0\\},$
$Y_{IV}:=Y_{IV}(N,n;K):=\\{W\in{\bf{C}^{N}},Z\in R_{IV}(n):$
$|W|^{2K}<1-2Z\overline{Z}^{t}+|ZZ^{t}|^{2},K>0\\},$

where $\det$ indicates the determinant, $N,m,n,p,q$ are natural numbers. These
domains are also called super-Cartan domains.

If the right hand of above inequalities are denoted by the
$N_{j}:=N_{j}(Z,\overline{Z}),j=I,II,III,IV$ respectively, then the definition
of Cartan-Hartogs domain can be also written as

$Y_{j}=\\{W\in{\bf{C}^{N}},Z\in
R_{j}:|W|^{2K}<N_{j}(Z,\overline{Z})\\},j=I,II,III,IV.$

The following topics will be discussed:

I. The zeroes of Bergman kernel of Cartan-Hartogs domain;

II. The classical (Cannonical) metrics (Bergman metric, Caratheodory metric,
Kaehler-Einstein metric, Kobayashi metric) on Cartan-Hartogs domain, which
contains Bergman metric equivalent to Kaehler-Einstein metric, Lu Qikeng
constant, Bergman Kaehler-Einstein metric and some good new metrics.

III. Generalized Cartan-Hartogs domain;

IV. The centre of representative domain and applications;

V. The solution of Dirichlet’s problem of complex Monge-Ampère equation on
Cartan-Hartogs domain and Kaehler-Einstein metric with explicit formula.

## I. The zeroes of Bergman kernel on Cartan-Hartogs domain

The Cartan-Hartogs domain of the first type is defined by

$Y_{I}(N,m,n;K)=\\{W\in{\bf{C^{N}}},Z\in
R_{I}(m,n):|W|^{2K}<det(I-Z\overline{Z}^{t}),K>0\\}.$

And

$Y_{I}(1,1,n;K)=\\{W\in{\bf{C}},Z\in{\bf{C^{n}}}:|W|^{2K}+|z_{1}|^{2}+|z_{2}|^{2}+\dots+|z_{n}|^{2}<1\\}.$

Then the Bergman kernel of $Y_{I}(1,1,n;K)$ is

$K_{I}((W,Z);(\zeta,\xi))=K^{-n}\pi^{-(n+1)}F(Y)det(I-Z\overline{\xi}^{t})^{-(1+n+1/K)}.$

Where

$F(Y)=\sum_{i=0}^{n+1}b_{i}\Gamma(i+1)Y^{i+1},Y=(1-X)^{-1},X=W\zeta[det(I-Z\overline{\xi}^{t})]^{-1/K}.$

And $b_{0}=0$, let

$P(x)=(x+1)[(x+1+Kn)(x+1+K(n-1))\dots(x+1+K)].$

Then the others $b_{i}(i=1,2,\dots\dots,n+1)$ can be got by

$None$
$b_{i}=[P(-i-1)-\sum_{k=0}^{i-1}b_{k}(-1)^{k}\Gamma(i+1)/\Gamma(i-k+1)][(-1)^{i}\Gamma(i+1)]^{-1}.$

Recently, Liyou Zhang prove that above formula can be rewritten as

$b_{i}=\sum_{j=1}^{i}\frac{(-1)^{j}P(-j-1)}{\Gamma(j+1)\Gamma(i-j+1)}.$

It is well known that for the first type of Cartan-Hartogs domains there
exists the holomorphic automorphism $(W^{*},Z^{*})=F(W,Z)$ such that
$F(W,Z_{0})=(W^{*},0)$ if $(W,Z_{0})\in Y_{I}$. Due to the transformation rule
of Bergman kernel, one has

$K_{I}((W,Z);(\zeta,\xi))=(detJ_{F}(W,Z))|_{Z_{0}=Z}K_{I}[(W^{*},0),(\zeta^{*},\xi^{*})](det\overline{J_{F}(\zeta,\xi)}).$

Therefore the zeroes of $K_{I}((W,Z);(\zeta,\xi))$ are same as the zeroes of
$K_{I}[(W^{*},0),(\zeta^{*},\xi^{*})]=K^{-n}\pi^{-(n+1)}F(Y)$. Let $W^{*}$ be
the $W$, and $\zeta^{*}$ be the $\zeta$, then we have

$None$
$K^{-n}\pi^{-(n+1)}F(Y)=K^{-n}\pi^{-(n+1)}F(y),y=(1-W\overline{\zeta})^{-1}.$

Where

$None$ $F(y)=\sum_{i=0}^{n+1}b_{i}\Gamma(i+1)y^{i+1}.$

If $(W,0)$, $(\zeta,0)$, $(W^{*},0)$ and $(\zeta^{*},0)$ belong to $Y_{I}$,
then their norms $|W|,|\zeta|,|W^{*}|,|\zeta^{*}|$ are less than 1.

1.1. Let $t=W\overline{\zeta}$, then $|t|<1$, and $F(y)=(1-t)^{-(n+2)}G(t)$,
where $G(t)=\sum_{i=0}^{(n+1)}b_{i}\Gamma(i+1)(1-t)^{n+1-i}$. Therefore to
discuss the the presence or absence of zeroes of the Bergman kernel function
of $Y_{I}(1,1,n;K)$ can be reduced to discuss the zeroes of polynomial with
real coefficients in the unit disk in $\bf{C}$[3].

Because $y=(1-W\overline{\zeta})^{-1}=\frac{1}{1-t}$, which maps the unit disk
in $t$-plane onto the half-plane in $y$-plane $Rey>1/2$, therefore to discuss
the the presence or absence of zeroes of the Bergman kernel function of
$Y_{I}(1,1,n;K)$ can be reduced to discuss the zeroes of polynomial with real
coefficients in the right half-plane $Rey>1/2$.

Above two statements are true not only for the $Y_{I}(1,1,n;K)$ but also for
all of the Cartan-Hartogs domains(and Hua domains).

1.2. In the low dimension case, it is very easy to answer the Lu Qikeng
problem for the Cartan-Hartogs domain. For example, we can say that
$Y_{I}(1,1,1;K)$ is Lu Qi-Keng domain.

At this time
$Y_{I}(1,1,1;K)=\\{W\in{\bf{C}},Z\in{\bf{C}}:|W|^{2K}+|Z|^{2}<1\\}$, and the
zeroes of its Bergman kernel function $K_{I}[(W,Z),(\zeta,\xi)]$ are same as
the zeroes of $K_{I}[(W^{*},0),(\zeta^{*},0)]$. But

$K_{I}[(W,0),(\zeta,0)]=K^{-n}\pi^{-(2)}F(y),F(y)=\sum_{i=0}^{2}b_{i}\Gamma(i+1)y^{i+1}$
$y=(1-t)^{-1}.$

Where $b_{1}=K-1,b_{2}=1,b_{0}=0$, therefore

$F(y)=(K-1)y^{2}+2y^{3}=y^{3}[(K-1)(1-t)+2].$

But the zeroes of $F(y)$ are equal to $t=(K+1)/(K-1)$, its norm $|t|>1$, it is
impossible.

Therefore the Bergman kernel function of $Y_{I}(1,1,1;K)$ is zero-free, that
is the $Y_{I}(1,1,1;K)$ is Lu Qi-Keng domain. Therefore we also prove that:

If $D\subset{C^{2}}$ is a bounded pseudoconvex domain with real analytic
boundary and its holomorphic automorphism group is noncompact, then $D$ is the
Lu Qi-Keng domain due to the E.Bedford and S.I.Pinchuk’s following theorem[4].

Theorem 1.1: If $D$ is a bounded pseudoconvex domain with real analytic
boundary and its holomorphic automorphism group is noncompact, then $D$ is
biholomorphically equivalent to a domain of the form

$E_{m}=\\{(z_{1},z_{2})\in{\bf{C^{2}}}:|z_{1}|^{2m}+|z_{2}|^{2}<1\\}$

for some positive integer $m$.

## II. The classical (Cannonical) metrics on Cartan-Hartogs domain

Let $D$ be the bounded domain in $\bf{C^{M}}$, $B_{D},C_{D},KE_{D},K_{D}$
denote the Bergman metric, Caratheodory metric, Kaehler-Einstein metric,
Kobayashi metric respectively, then we have $C_{D}\leq B_{D},C_{D}\leq K_{D}$
[5], and if $D$ is also the convex domain then $C_{D}=K_{D}$[5]. On the other
hand, there is no clear relationship between the $B_{D}$ and $K_{D}$.

2.1. But we have that $B_{D}\leq cK_{D}$ for the Cartan-Hartogs domain where
$c$ is the constant [6-9].

2.2. We proved the Bergman metric is equivalent to Kaehler-Einstein
metric[10], that is $B_{D}\sim KE_{D}$ on Cartan-Hartogs domains. For example,
we consider the Cartan-Hartogs of the first type $Y_{I}=Y_{I}(N,m,n;K)$. Let

$G_{\lambda}=G_{\lambda}(Z,W)=Y^{\lambda}[\det(I-Z\overline{Z}^{t})]^{-(m+n+\frac{N}{K})},\lambda>0,$
$T_{\lambda
I}(Z,W;\overline{Z},\overline{W})=(g_{i\overline{j}})=\left(\frac{\partial^{2}\log
G_{\lambda}}{\partial z_{i}\partial\overline{z}_{j}}\right),$

where

$Y=(1-X)^{-1},X=|W|^{2}[\det(I-Z\overline{Z}^{t})]^{-\frac{1}{K}}.$

Then $G_{\lambda}$ induces a metric

$Y(I\lambda):=[(dw,dz)T_{\lambda
I}(Z,W;\overline{Z},\overline{W})\overline{(dw,dz)}^{t}]^{1/2}.$

Firstly, by the direct computations one can prove that $B_{Y_{I}}\sim
Y(I\lambda)$. The $Y(I\lambda)$ has good properties: Its holomorphic sectional
curvature and Ricci curvature are bounded from above and below by the Negative
constants. Then based on above good properties and using the Yau’s Schwarz
lemma[11] one can prove $KE_{Y_{I}}\sim Y(I\lambda)$. Therefore
$KE_{Y_{I}}\sim B_{Y_{I}}$, and the metric $Y(I\lambda)$ may be useful for us.

2.3. Definition: A complex manifold $M^{n}$ is called holomorphic homogeneous
regular if there are positive constants $r<R$ such that for each point $p\in
M$, there is a one to one holomorphic map $f:M\longrightarrow\bf{C}^{n}$ such
that

i) $f(p)=0;$

ii) $B_{r}\subset f(M)\subset B_{R},$ where $B_{r}$ and $B_{R}$ are balls with
radius $r$ and $R$ respectively.

Theorem 2.1(Liu-Sun-Yau)[11,12]: For holomorphic homogeneous regular
manifolds, the Bergman metric, the Kobayashi metric and the Caratheodory
metric are equivalent.

Therefore if Cartan-Hartogs domains are the holomorphic homogeneous regular
manifolds, then the Bergman metric, the Kobayashi metric, the Caratheodory
metric and Kaehler-Einstein metric are equivalent. But whether the Cartan-
Hartogs domains are the holomorphic homogeneous regular manifolds? This
problem remains open.

2.4. From an immediate consequence of an inequality due to Lu Qikeng’s paper
[13], we have the following

Theorem 2.2(Lu Qikeng): Let $D$ be a bounded domain in $\bf{C}^{n}$. Then for
each tangent $v\in T_{z}(D)=\bf{C}^{n}$ at $z\in D$, $B_{D}(z,v)\geq
C_{D}(z,v)$. Where $B_{D}(z,v)$ equals the length of $v$ w.r.t. the Bergman
metric $B_{D}$, and $C_{D}(z,v)$ equals the length of $v$ w.r.t. the
differential Caratheodory metric $C_{D}$.

Therefore Cheung and Wong introduce the definition of Lu constant $L(D)$ of a
bounded domain $D$ in $\bf{C}^{n}$ as follows[5].

Definition:

$L(D)=\sup\limits^{z\in
D}_{v\not=0\in{T_{z}(D)}}(\frac{C_{D}(z,v)}{B_{D}(z,v)})$

Lu’s theorem says that $L(D)\leq 1$. $L(D)=(1/(n+1))^{1/2}$ when $D$ is the
unit ball in $\bf{C}^{n}$. One can try to determine the Lu’s constants of all
Cartan domains and all Cartan-Hartogs domains.

2.5. Some years ago S.T.Yau proposed an intricate problem to look for a
characterization of the bounded pseudoconvex domains on which the Bergman
metrics are complete Kaehler-Einstein metric[5].

The following Lu’s theorem can be viewed as a particular case of Yau’s problem
of which the Bergman metric is of constant negative holomorphic sectional
curvature:

Theorem 2.3(Lu Qikeng)[14]: Let $D$ be a bounded domain in $\bf{C}^{n}$ with a
complete Bergman metric $B_{D}$. If the holomorphic sectional curvature is
equal to a negative constant $-c^{2}$, then $D$ is biholomorphic to the
Euclidean ball $B_{n}$ and $c^{2}=\frac{2}{n+1}$.

S.Y.Cheng conjectures that a strangle pseudoconvex domain whose Bergman metric
is Kaehler-Einstein must be biholomorphic to the Euclidean ball[5].

We can also prove that if the Bergman metric of Cartan-Hartogs domain is
Kaehler-Einstein, then this Cartan-Hartogs domain must be homogeneous(See
below).

## III. Generalized Cartan-Hartogs domain

Some years ago we generate the Cartan-Hartogs domain to the Hua domain as
follows[3]:

$\begin{array}[]{lll}\\{W_{j}\in{\bf{C}^{N_{j}}},Z\in{R_{s}}:\displaystyle\sum_{j=1}^{r}\frac{||W_{j}||^{2p_{j}}}{[N_{s}(Z,\overline{Z})]^{K_{j}}}<1,\\\
p_{j}>0,K_{j}>0,j=1,\dots,r\\}.s=I,II,III,IV.\end{array}$

Right now we will generate the Cartan-Hartogs domain from another way.

Let $\Omega$ be a domain in $\bf{C}^{n}$, $\rho$ a positive continuous
function on $\Omega$, and let $D$ be a (fixed) irreducible bounded symmetric
domain in $\bf{C}^{d}$. Then Roos, Engli$\breve{s}$ and Zhang define a new
domain in $\bf{C}^{n+d}$ as follows[15,16]:

$None$ $\Omega^{D}:=\\{(w,z)\in{\bf{C}^{d}\times\Omega}:\frac{w}{\rho(z)}\in
D\\}.$

Let $B(0,1)$ be the unit ball in $\bf{C}^{d}$, and let

$D=B(0,1),\rho(z)=N_{j}(z,z)^{1/(2K)},$

then one has

$None$
$\Omega^{B(0,1)}:=\\{(w,z)\in{\bf{C}^{d}\times\Omega}:\frac{w}{N_{j}(z,z)^{1/(2K)}}\in
B(0,1)\\}.$

The $\frac{w}{N_{j}(z,z)^{1/(2K)}}\in B(0,1)$ can be denoted by

$(\frac{w}{N_{j}(z,z)^{1/(2K)}})\overline{(\frac{w}{N_{j}(z,z)^{1/(2K)}})}^{\prime}<1.$

That is

$|w|^{2K}<N_{j}(z,z).$

Therefore

$None$
$\Omega^{B(0,1)}=\\{(w,z)\in{\bf{C}^{d}\times\Omega}:|w|^{2K}<N_{j}(z,z)\\}.$

Above (3.3) is the definition of Cartan-Hartogs domain.

Let

$D=R_{j},\rho(z)=N_{i}(z,z)^{1/(2K)},d=dimD,\Omega=R_{i},$

then we get the following new domain, which generalizes the Cartan-Hartogs,
and is called generalized Cartan-Hartogs domain:

$None$ $R_{i}^{R_{j}}=\\{(w,z)\in{\bf{C}^{d}\times
R_{i}}:\frac{w}{N_{i}(z,z)^{1/(2K)}}\in R_{j}\\}.$

where $i,j=I,II,III,IV$. Therefore we get 16 types of generalized Cartan-
Hartogs domain as follows:

$Y(I,I)=R_{I}^{R_{I}}=\\{(w,z)\in{\bf{C}^{mn}\times
R_{I}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(m)}\\}.$
$Y(I,II)=R_{I}^{R_{II}}=\\{(w,z)\in{\bf{C}^{p(p+1)/2}\times
R_{I}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(p(p+1)/2)}\\}.$
$Y(I,III):=R_{I}^{R_{III}}=\\{(w,z)\in{\bf{C}^{q(q-1)/2}\times
R_{I}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(q(q-1)/2)}\\}.$
$Y(I,IV):=R_{I}^{R_{IV}}=\\{(w,z)\in{\bf{C}^{N}\times
R_{I}}:2det(I-z\overline{z^{\prime}})^{1/(K)}w\overline{w^{\prime}}-|ww^{\prime}|^{2}<det(I-z\overline{z^{\prime}})^{2/(K)},$
$|ww^{\prime}|<det(I-z\overline{z^{\prime}})^{1/(K)}\\}.$
$Y(II,I):=R_{II}^{R_{I}}=\\{(w,z)\in{\bf{C}^{mn}\times
R_{II}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(m)}\\}.$
$Y(II,II):=R_{II}^{R_{II}}=\\{(w,z)\in{\bf{C}^{p(p+1)/2}\times
R_{II}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(p)}\\}.$
$Y(II,III):=R_{II}^{R_{III}}=\\{(w,z)\in{\bf{C}^{q(q-1)/2}\times
R_{II}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(q)}\\}.$
$Y(II,IV):=R_{II}^{R_{IV}}=\\{(w,z)\in{\bf{C}^{N}\times
R_{II}}:2det(I-z\overline{z^{\prime}})^{1/(K)}w\overline{w^{\prime}}-|ww^{\prime}|^{2}<det(I-z\overline{z^{\prime}})^{2/(K)},$
$|ww^{\prime}|<det(I-z\overline{z^{\prime}})^{1/(K)}\\}.$
$Y(III,I):=R_{III}^{R_{I}}=\\{(w,z)\in{\bf{C}^{mn}\times
R_{III}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(m)}\\}.$
$Y(III,II):=R_{III}^{R_{II}}=\\{(w,z)\in{\bf{C}^{p(p+1)/2}\times
R_{III}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(p)}\\}.$
$Y(III,III):=R_{III}^{R_{III}}=\\{(w,z)\in{\bf{C}^{q(q-1)/2}\times
R_{III}}:w\overline{w^{\prime}}<det(I-z\overline{z^{\prime}})^{1/(K)}I^{(q)}\\}.$
$Y(III,IV):=R_{III}^{R_{IV}}=\\{(w,z)\in{\bf{C}^{N}\times
R_{III}}:2det(I-z\overline{z^{\prime}})^{1/(K)}w\overline{w^{\prime}}-|ww^{\prime}|^{2}<det(I-z\overline{z^{\prime}})^{2/(K)},$
$|ww^{\prime}|<det(I-z\overline{z^{\prime}})^{1/(K)}\\}.$
$Y(IV,I):=R_{IV}^{R_{I}}=\\{(w,z)\in{\bf{C}^{mn}\times
R_{IV}}:w\overline{w^{\prime}}<(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{1/(K)}I^{(m)}\\}.$
$Y(IV,II):=R_{IV}^{R_{II}}=\\{(w,z)\in{\bf{C}^{p(p+1)/2}\times
R_{IV}}:w\overline{w^{\prime}}<(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{1/(K)}I^{(p)}\\}.$
$Y(IV,III):=R_{IV}^{R_{III}}=\\{(w,z)\in{\bf{C}^{q(q-1)/2}\times
R_{IV}}:w\overline{w^{\prime}}<(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{1/(K)}I^{(q)}\\}.$
$Y(IV,IV):=R_{IV}^{R_{IV}}=\\{(w,z)\in{\bf{C}^{N}\times
R_{IV}}:2(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{1/(K)}w\overline{w^{\prime}}-|ww^{\prime}|^{2}<(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{2/(K)},$
$|ww^{\prime}|<(1+|zz^{\prime}|^{2}-2z\overline{z^{\prime}})^{1/(K)}\\}.$

These are the new research fields, one can seeks the Bergman kernel,
Szeg$\ddot{o}$ and consider other topics.

## IV. The centre of representative domain and applications

The Riemann mapping theorem characterizes the planar domains that are
biholomorphically equivalent to the unit disk. In the higher dimensions, there
is no Riemann mapping theorem, and the following problem arise:

Are there canonical representatives of biholomorphic equivalence classes of
domains?

In the dimension one, if $K(z,w)$ is the Bergman kernel function of simply
connected domain $D\neq{\bf{C}}$, it is well known that the biholomorphic
mapping

$F(z)=\frac{1}{K(t,t)}\frac{\partial}{\partial\overline{w}}log\frac{K(z,w)}{K(w,w)}|_{w=t}$

maps the $D$ onto unit disk.

In the higher dimensions, Stefan Bergman introduced the notion of a
”representative domain” to which a given domain may be mapped by
”representative coordinates”. If $D$ is a bounded domain in ${\bf{C^{n}}}$,
$K(Z,W)$ is the Bergman kernel function of $D$, let

$T(Z,Z)=(g_{ij})=(\frac{\partial^{2}logK(Z,W)}{\partial
z_{i}\partial\overline{z_{j}}})$

and its converse is $T^{-1}(Z,W)=(g^{-1}_{ji})$. Then the local representative
coordinates based at the point $t$ is

$f_{i}(Z)=\sum^{n}_{j=1}g^{-1}_{ji}\frac{\partial}{\partial{\overline{W_{j}}}}log\frac{K(Z,W)}{K(W,W)}\mid_{W=t},i=1,\dots,n.$

Or

$F(Z)=(f_{1},\dots,f_{n})=\frac{\partial}{\partial{\overline{W}}}log\frac{K(Z,W)}{K(W,W)}\mid_{W=t}T^{-1}(t,t),$

where

$\frac{\partial}{\partial{\overline{W}}}=(\frac{\partial}{\partial{\overline{W_{1}}}},\frac{\partial}{\partial{\overline{W_{2}}}},\dots,\frac{\partial}{\partial{\overline{W_{n}}}}).$

These coordinates take $t$ to $0$ and have complex Jacobian matrix at $t$ is
equal to the identity. The $F(D)$ is called the representative domain of $D$.
If $D$ is biholomorphic equivalent to $D_{1}$, then $D$ and $D_{1}$ have same
representative domain.

Zeroes of the Bergman kernel function $K(Z,W)$ evidently pose an obstruction
to the global definition of Bergman representative coordinates. This
observation was Lu Qi-Keng’s motivation for asking which domains have zero-
free Bergman kernel functions. This problem is called Lu Qi-Keng conjecture by
M.Skwarczynski in 1969 in his paper [17]. If the Bergman kernel function of
$D$ is zero-free, that means the Lu Qi-Keng conjecture has a positive answer,
then the domain $D$ is called the Lu Qi-Keng domain.

4.1. In 1981 Lu Qikeng introduces an another definition of ”representative
domain”[18].

Definition: A bounded domain in $\bf{C}^{n}$ is called a representative
domain, if there is a point $t\in{D}$ such that the matrix of the Bergman
metric tensor $T(z,\overline{t})$ is independent of $z\in{D}$. The point $t$
is called the centre of the representative domain.

If $D$ is representative domain in the sense of Lu, and $D_{1}$ is the
representative domain of $D$ in the sense of Bergman, then $D$ is same as the
$D_{1}$ under an affine transformation.

4.2. In 1981, Lu Qikeng[18] proved the following

Theorem 4.1: Let $D$ be a bounded domain and $D_{1}$ be a representative
domain of Lu in $\bf{C}^{n}$ with centre $s_{0}$. If $f:D\longrightarrow
D_{1}$ is a biholomorphic mapping, then $f$ is of the form

$f(z)=s_{0}+[\frac{\partial}{\partial\overline{t}}log\frac{K(z,\overline{t})}{K(t,\overline{t})}]_{t=t_{0}}T^{-1}(t_{0},\overline{t_{0}})A.$

Moreover $K(z,\overline{t_{0}})$ is zero free when $z\in D$. Where
$s_{0}=f(t_{0})$, $A=(\frac{\partial f}{\partial z})_{z=t_{0}}$, and
$\frac{\partial}{\partial\overline{t}}=(\frac{\partial}{\partial\overline{t_{1}}},\dots,\frac{\partial}{\partial\overline{t_{n}}})$.

Corollary 1: Let $D$ be a bounded domain and $D_{1}$ be a representative
domain of Lu in $\bf{C}^{n}$ with centre $s_{0}$, if $D$ is biholomorphic
equivalent $D_{1}$, the Bergman kernel of $D_{1}$ is $K_{1}(w,\overline{s})$,
then $K_{1}(w,\overline{s_{0}})$ is zero free when $w\in{D_{1}}$.

Corollary 2: If $s_{0}=0$, $D=D_{1}$, then $A=I$, and the holomorphic
automorphism $f(z)$ of $D$ has the following form:

$f(z)=[\frac{\partial}{\partial\overline{t}}log\frac{K(z,\overline{t})}{K(t,\overline{t})}]_{t=t_{0}}T^{-1}(t_{0},\overline{t_{0}}).$

Where $0=f(t_{0})$.

4.3. If the full group of holomorphic automorphism is denoted by $Aut(D)$, and
let $S=\\{z:f(z)=0,f\in Aut(D)\\}$. Then the $Aut(D)$ is constituted by

$f(z)=[\frac{\partial}{\partial\overline{t}}log\frac{K(z,\overline{t})}{K(t,\overline{t})}]_{t=t_{0}}T^{-1}(t_{0},\overline{t_{0}}).$

Where $t_{0}$ spreads all over $S$.

Therefore if $0$ is the centre of representative domain $D$, and the set $S$
is got explicitly, then the $Aut(D)$ can be got explicitly as above. From
this, we can get the $Aut(D)$ with explicit form if $D$ is the Cartan-Hartogs
domain.

## V. The solution of Dirichlet’s problem of complex Monge-Ampère equation on
Cartan-Hartogs domain and Kaehler-Einstein metric with explicit formula

Complex Monge-Ampère equation is the nonlinear equation with high degree,
therefore to get its solution is very difficult.

S.Y.Cheng, N.M. Mok, S.T. Yau consider the following Dirichlet’s problem of
the complex Monge-Ampère equation:

$\left\\{\begin{array}[]{ll}\displaystyle{\rm{det}}\left(\frac{\partial^{2}g}{\partial
z_{i}\partial\overline{z}_{j}}\right)=e^{(n+1)g}&z\in D,\\\ \displaystyle
g=\infty&z\in\partial D,\\\ \end{array}\right.$

And they proved that the above problem exists unique solution[19,20], where
the $g$ can induce the Kaehler-Einstein metric as follows:

$(KE_{D})^{2}=dz(\frac{\partial^{2}g}{\partial
z_{i}\partial\overline{z}_{j}})\overline{dz}^{t}.$

We consider the explicit solution of Dirichlet’s problem of complex Monge-
Ampère equation on $Y_{I}$:

$None$
$\left\\{\begin{array}[]{ll}\displaystyle{\rm{det}}\left(\frac{\partial^{2}g}{\partial
z_{i}\partial\overline{z}_{j}}\right)_{1\leq i,j\leq M}=e^{(M+1)g}&z\in
Y_{I},\\\ \displaystyle g=\infty&z\in\partial Y_{I},\\\ \end{array}\right.$

where $M=N+mn$ is the complex dimension of $Y_{I}$.

Because $Y_{I}$ is pseudoconvex domain. Therefore the solution of problem
(5.1) is existent and unique.

5.1. We prove that the solution of problem (5.1) can be got in semi-explicit
formula, and the explicit solution is obtained in special case. That is the
following theorem is proved[21]:

Theorem 5.1: If $G(X)$ is the solution of the following problem

$None$
$\left\\{\begin{array}[]{lll}(M+1)^{-M}[\frac{X}{K}G^{\prime}+(m+n+\frac{N}{K})G]^{mn}[GG^{\prime}+(GG^{\prime\prime}-(G^{\prime})^{2})X]\frac{(G^{\prime})^{N-1}}{G^{M+1}}=G,\\\
G(0)=\displaystyle K^{-mn};\displaystyle lim_{X\rightarrow
1}G(X)=\infty,\end{array}\right.$

then

$g=(M+1)^{-1}log[G(X)det(I-Z\overline{Z}^{t})^{-(m+n+N/K)}]$

is the solution of the problem (5.1); if $K=\frac{mn+1}{m+n}$, and

$G(X)=(\frac{m+n}{mn+1})^{mn}(1-X)^{-(M+1)},$

then the following $g$ is the special solution of the problem (5.1):

$g=(M+1)^{-1}log[(\frac{m+n}{mn+1})^{mn}(1-X)^{-(M+1)}det(I-Z\overline{Z}^{t})^{-(m+n+N/K)}]$
$None$
$=log[(1-X)^{-1}det(I-Z\overline{Z}^{t})^{-\frac{m+n}{mn+1}}(\frac{m+n}{mn+1})^{\frac{mn}{M+1}}],$

where

$X=X(Z,W)=|W|^{2}[det(I-Z\overline{Z}^{t})]^{-1/K},G^{\prime}=\frac{dG(X)}{dX},G^{\prime\prime}=\frac{d^{2}G(X)}{dX^{2}}.$

Remark 1: The complex Monge-Ampère equation is the nonlinear equation, hence
to get its explicit solution is very difficult. Therefore mathematicians hope
to get the solutions for the problem (5.1) by using the numerical method. Due
to the above results the numerical method of the problem (5.1) is reduced to
the numerical method of the problem (5.2). Which reduce the complexity of the
numerical method of problem (5.1) consumedly. Next, if the numerical method of
problem (5.1) or the numerical method of problem (5.2) is appeared in the
future, then the special solution $g$ (see (5.3)) can be used to check these
numerical methods. And if one reduces the complex Monge-Ampère equation in
(5.1) by the linearization method, then the $g$ of (5.3) can be also to check
the precision and the rationality for the linearization method.

Remark 2: Although the problem (5.1) have not been got the explicit solution
in general case, but its semi-explicit solution has the form

$g=(M+1)^{-1}log[G(X)det(I-Z\overline{Z}^{t})^{-(mn+N/K)}],$

where $G(X)$ satisfies the (5.2).

Remark 3: The Bergman kernel function of $Y_{I}(N,m,n;K)$ is

$K_{I}(W,Z;\overline{W},\overline{Z})=K^{-mn}\pi^{-(mn+N)}G(X)det(I-Z\overline{Z}^{t})^{-(m+n+N/K)}.$

Where

$G(X)=\sum_{i=0}^{mn+1}b_{i}\Gamma(N+i)(1-X)^{-(N+i)},X=X(W,Z)=|W|^{2}[det(I-Z\overline{Z}^{t})]^{-1/K},$
$|W|^{2}=\sum_{j=1}^{N}|W_{j}|^{2},$

and $b_{i}$ are constants.

If $Y_{I}$ is homogeneous domain, then its Bergman metric is equal to its
Kaehler-Einstein metric, that is
$G(X)=\sum_{i=0}^{mn+1}b_{i}\Gamma(N+i)(1-X)^{-(N+i)}$ must satisfies the
equation (5.2). If $G(X)$ is not satisfies (5.2), then $Y_{I}$ is not
homogeneous.

If $G(X)$ is satisfies (5.2), then the Bergman metric of $Y_{I}$ is equal to
the Kaehler-Einstein metric.

By computations, we prove that the $G(X)$ satisfies the equation (5.2) if and
only if $m=1$.That is the $Y_{I}$ is the unit ball(homogeneous domain).

## References

  * [1] HUA L K. Harmonic Analysis of Functions of Several Complex Variables in Classical Domains. Providence: Amer.Math.Soc., 1963.
  * [2] YIN Weiping. The Bergman kernels on super-Cartan domain of the first type, Science in China(series A), 2000, 43(1): 13-21.
  * [3] YIN Weiping. Lu Qikeng conjecture and Hua domain. Science in China Series A: Mathematics, 2008, 51(4): 803-818
  * [4] BEDFORD E., PINCHUK S.I.. Domains in $\bf{C^{2}}$ with noncompact group of automorphism. (Russian) Mat.Sb., 1988, 135: 147-157; (English) Math. USSR Sbornik, 1989, 63:141-151.
  * [5] CHEUNG Wing-Sum, WONG Bun. Remarks on two theorems of LU Qikeng. Science in China Series A: Mathematics, 2008, 51(4):773-776.
  * [6] YIN Weiping, WANG An, ZHAO Xiaoxia. The comparison theorem for the Bergman and Kobayashi metrics on Cartan-Hartogs domains of the first type. Science in China (Series A), 2001, 44(5): 587-598.
  * [7] ZHAO Xiaoxia, DING Li, YIN Weiping. The comparison theorem on Cartan-Hartogs domain of the second type. Progress in Natural Science, 2004, 14(2): 105-112.
  * [8] YIN Weiping, ZHAO Xiaoxia. The comparison theorem on Cartan-Hartogs domain of the third type. Complex Variables, 2002, 47(3): 183-201.
  * [9] LIN Ping, YIN Weiping. The comparison theorem on Cartan-Hartogs domain of the fourth type(In Chinese). Advanced in Mathematics(CHINA), 2003,32(6):739-750.
  * [10] YIN Weiping, WANG An. The equivalence on classical metrics, Science in China Series A: Mathematics, 2007, 50(2): 183-200.
  * [11] LIU Kefeng, SUN Xiaofeng, YAU Shing-Tung. Geometric aspects of the moduli space of Riemann surfaces. Science in China Ser.A Mathematics, 2005, 48(Supp): 97-122.
  * [12] YAU Shing-Tung. Canonical metrics on complex manifolds. Science in China Series A: Mathematics, 2008, 51(4): 503-508.
  * [13] LU Qikeng. Schwarz lemma and analytic invariants. Scientia Sinica, 1957,7: 453-504.
  * [14] LU Qi-Keng. On Kähler manifolds with constant curvature(in Chinese). Acta Mathematica Sinica, 1966, 16:269-281. Translated as ”Chinese Mathematics”, 1966,8:283-298.
  * [15] ROOS Guy. Weighted Bergman kernels and virtual Bergman kernels. Science in China Ser.A Mathematics, 2005, 48(Supp.): 225-237.
  * [16] ENGLI$\breve{S}$ Miroslav, ZHANG Genkai. On a generalized Forelli-Rudin construction. Complex Variables and Elliptic Equations, 2006, 51(3):277-294.
  * [17] M.Skwarczynski. The distance in theory of pseu-conformal transformations and the Lu Qi-Keng conjecture . Proc. Amer. Math. Soc., 1969, 22: 305-310.
  * [18] Lu Qikeng. On the representative domain. In Several Complex Variables Proceedings of the 1981 Hangzhou Conference, Eds: J.J.Kohn, Q.-K. Lu, R.Remmert, Y.-T. Siu, Boston: Birkh$\ddot{a}$user Boston, Inc., 1984.
  * [19] CHENG S.Y. and YAU S.T. On the existence of a complete Kähler metric on non-compact complex manifolds and the regularity of Fefferman’s equation. Comm. Pure App. Math., 1980, 33: 507-544.
  * [20] MOK N. and YAU S. T. Completeness of the Kähler-Einstein metric on bounded domain and the characterization of domain of holomorphy by curvature conditions. Proc Symposia Pure Math., 1983, 39: 41-59.
  * [21] YIN Weiping, YIN Xiaolan. On the solution of Dirichlet’s problem of complex Monge-Ampere equation for Cartan-Hartogs domain of the first type with Yin Xiaolan Nonlinear Analysis Series A Theory, Methods and Applications(to appear).