Bergman metrics as pull-backs of the Fubini-Study metric

TL;DR

This paper characterizes complex manifolds with constant holomorphic sectional curvature using Bergman metrics as pull-backs of the Fubini-Study metric, introduces new domains with unique curvature properties, and proposes a new conjecture.

Contribution

It presents a novel approach to understanding Bergman metrics via pull-backs of the Fubini-Study metric and constructs new domains with distinctive curvature characteristics.

Findings

Characterization of manifolds with constant holomorphic sectional curvature

Construction of new domains with surprising curvature properties

Formulation of a new conjecture regarding Bergman metrics

Abstract

Domains and more generally complex manifolds whose Bergman metrics have constant holomorphic sectional curvature are characterized. Our approach is to treat the Bergman metrics as the pull-back by the Bergman-Bochner maps of the Fubini-Study metric of the complex projective space of infinite dimension. Several new domains with surprising curvature properties for their Bergman metrics are constructed. A new conjecture is also formulated at the end of the paper.

Equations

$${\ell }^2=\{a=(a_1,\cdots ,a_m,\cdots ):\Vert a{\Vert }^2=:\Vert a{\Vert }_{{\ell }^2}^2:=\sum _{j=1}^{\infty }|a_j{|}^2<\infty ,a_j\in \mathbb{C}\}$$

$$\Vert F(z)-F_N(z){\Vert }_{{\ell }^2}^2\le \frac{C_1}{2^N}{\int }_0^{2\pi }\cdots {\int }_0^{2\pi }\Vert F(e^{i{\theta }_1},\cdots e^{i{\theta }_n}){\Vert }^{2}d{\theta }_1\cdots d{\theta }_n\le \frac{C_2}{2^N}.$$

$$\omega_{st}=i\partial\overline{\partial}\log\Big{(}\sum_{j=1}^{\infty}|z_{j}|^{2}\Big{)}.$$

$$[F]=[f_1,\cdots ,f_n,\cdots ].$$

$$\mathcal{F}:(M,\omega )\to ({\mathbb{P}}^{\infty },{\omega }_{st})$$

$$\mathcal{F}^{*}(\omega_{st})=i\partial\overline{\partial}\log\Big{(}\sum_{j=1}^{\infty}|f_{j}(z)|^{2}\Big{)}=\omega\quad\hbox{over }U,$$

$${\mathcal{F}}^{\ast }({\omega }_{st})={\mathcal{G}}^{\ast }({\omega }_{st}).$$

$$\mathcal{F}(z)=[F],F=(F_1,\cdots ,F_n,\cdots ),\mathcal{G}(z)=[G],G=(G_1,\cdots ,G_n,\cdots )$$

$$\Vert F(z){\Vert }^2=\sum _{j=1}^{\infty }|F_j(z){|}^{2}and\Vert G(z){\Vert }^2=\sum _{j=1}^{\infty }|G_j(z){|}^2$$

$$[F{]}^{\ast }({\omega }_{st})=[G{]}^{\ast }({\omega }_{st})⟺\partial \overline{\partial }\log (\Vert F(z){\Vert }^2)=\partial \overline{\partial }\log (\Vert G(z){\Vert }^2).$$

$$\log (1+\sum _{j=2}^{\infty }|F_j(z){|}^2)-\log (1+\sum _{j=2}^{\infty }|G_j(z){|}^2)=\Re h(z),$$

$$\Vert F(z){\Vert }^2=\Vert G(z){\Vert }^{2}z\in U.$$

$$⟨F(z),F(w)⟩=⟨G(z),G(w)⟩z,w\in U.$$

$$X_F^o=Span\{{\partial }^{\alpha }F(0)=\frac{{\partial }^{|\alpha |}F}{\partial z^{\alpha }}(0):\alpha =({\alpha }_1,\cdots ,{\alpha }_n)\in ({\mathbb{Z}}^{+}{)}^n\}\subset {\ell }^2$$

$$T({\partial }^{\alpha }F(0))={\partial }^{\alpha }G(0),\alpha \in {\mathbb{Z}}_{+}^n$$

$$⟨{\partial }^{\alpha }F(0),{\partial }^{\beta }F(0)⟩=⟨{\partial }^{\alpha }G(0),{\partial }^{\beta }G(0)⟩.$$

$$\sum _{j=1}^m{a}_j{\partial }^{{\alpha }_j}F(0)=0\; if\; and\; only\; if\sum _{j=1}^m{a}_j{\partial }^{{\alpha }_j}G(0)=0.$$

$$⟨\sum _{j=1}^m{a}_j{\partial }^{{\alpha }_j}G(0),\sum _{j=1}^m{b}_j{\partial }^{{\alpha }_j}G(0)⟩=⟨T(\sum _{j=1}^m{a}_j{\partial }^{{\alpha }_j}F(0)),T(\sum _{j=1}^m{b}_j{\partial }^{{\alpha }_j}F(0))⟩$$

$$=⟨\sum _{j=1}^m{a}_j{\partial }^{{\alpha }_j}F(0),\sum _{j=1}^m{b}_j{\partial }^{{\alpha }_j}F(0)⟩.$$

$$T(F(z))=\sum _{|\alpha |=0}^{\infty }\frac{T({\partial }^{\alpha }F(0))}{\alpha !}{z}^{\alpha }=\sum _{|\alpha |=0}^{\infty }\frac{{\partial }^{\alpha }G(0)}{\alpha !}{z}^{\alpha }=G(z)forznear\; 0.$$

$$\omega (z)=i\partial \overline{\partial }\log \varphi (z,\overline{z}),\varphi =e^{\psi }.$$

$$\varphi (z,\overline{z})=1+2\Re {\varphi }_0(z)+O((z-\gamma (t_0))\cdot \overline{(z-\gamma (t_0))})$$

$$\log \varphi (z,\overline{z})=h_0(z)+\overline{h_0(z)}+\log (1+\sum _{|\alpha |,|\beta |>0}{b}_{\alpha \overline{\beta }}(z-\gamma (t_0){)}^{\alpha }{\overline{(z-\gamma (t_0))}}^{\beta })$$

$$i\partial \overline{\partial }\log \varphi (z,\overline{z})=i\partial \overline{\partial }\log (1+\sum _{k=2}^{\infty }|f_k(z){|}^2),$$

$$\log\Big{(}1+\sum_{|\alpha|,|\beta|>0}b_{\alpha\overline{\beta}}(z-\gamma(t_{0}))^{\alpha}\overline{(z-\gamma(t_{0}))}^{\beta}\Big{)}-\log(1+\sum_{j=2}^{\infty}|f_{j}(z)|^{2})=0.$$

$$\sum _{|\alpha |,|\beta |>0}{b}_{\alpha \overline{\beta }}(z-\gamma (t_0){)}^{\alpha }{\overline{(z-\gamma (t_0))}}^{\beta }=\sum _{k=2}^{\infty }{f}_k(z)\overline{f_k(z)}forznear\gamma (t_0).$$

$$b_{\alpha\overline{\alpha}}=\sum_{k=2}^{\infty}\Big{|}{1\over\alpha!}{\partial^{\alpha}f_{k}(0)}\Big{|}^{2}.$$

$$|b_{\alpha \overline{\alpha }}|\le C_1{{\delta }_1}^{-2|\alpha |}$$

$$\Big{|}{1\over\alpha!}{\partial^{\alpha}f_{k}(0)}\Big{|}^{2}\leq|b_{\alpha\overline{\alpha}}|\leq C_{1}\delta_{1}^{-2|\alpha|},\ \hbox{or }\ \Big{|}{1\over\alpha!}{\partial^{\alpha}f_{k}(0)}\Big{|}\leq\sqrt{|b_{\alpha\overline{\alpha}}|}\leq\sqrt{C_{1}}\delta_{1}^{-|\alpha|}.$$

$$f_k(z)=\sum _{|\alpha |=1}^{\infty }\frac{{\partial }^{\alpha }{f}_k(0)}{\alpha !}{z}^{\alpha }$$

Full text

Bergman metrics as pull-backs of the Fubini-Study metric

Xiaojun Huang111Supported in part by DMS-2000050   Song-Ying Li

( )

1 Introduction

In this paper, we study curvature properties of Bergman metrics of domains in ${\mathbb{C}}^{n}$, and more generally, of complex manifolds. There are classical results due to S. Bergman [Ber], S. Kobayashi [Ko] and Q. Lu [Lu], etc. Bergman showed in [Ber] that holomorphic sectional curvatures of the Bergman metric of a bounded domain in ${\mathbb{C}}^{n}$ is bounded by the one for projective spaces, i.e, is bounded from above by $+2$. Kobayshi generalized it to complex manifolds in [Ko]. Qi-Keng Lu proved in [Lu] that a domain $\Omega$ in ${\mathbb{C}}^{n}$ with a complete Bergman metric (thus $\Omega$ is pseudoconvex [Bre]) has a constant holomorphic sectional curvature if and only if $\Omega$ is biholomorphic to the ball. In [FW], [NS1] and [HX2], it is shown that a smoothly bounded strongly pseudo-convex domain in ${\mathbb{C}}^{n}$ has its Bergman metric Einstein if and only if it is biholomorphic to the ball as conjectured by S. Y. Cheng in [Ch]. There are also works (e.g, [Kle]) studying the asymptotic behavior of holomorphic sectional curvatures near a strongly pseudoconvex point, using Fefferman’s asymptotic expansion for Bergman kernel functions [Fe].

Since many pseudoconvex domains in ${\mathbb{C}}^{n}$ do not have complete Bergman metrics, after the work of Lu [Lu], the following has become a natural question: How to characterize a pseudoconvex domain with an incomplete negative constant holomorphic sectional Bergman curvature ? In two recent papers by X. Dong and B. Wong [DW1] and [DW2], the authors made an important progress on this problem. Applying the Bergman coordinate maps, they proved that such a domain $D$ is biholomorphic to the unit ball with a pluripolar set removed under the assumption that the Calabi diastasis function $K(z,z)|K(z,p)|^{-2}$ blows up on the boundary of the domain in [DW1], where $K(z,w)$ is the Bergman kernel function of the domain $D$ with $z,w\in D$. In [DW2], when $D$ is a planar domain with Gaussian curvature identically equal to $-2$, they proved that if any Dirichlet regular boundary point $q\in\partial D$ is the endpoint of a simple arc in $D$ lying in the exterior of $D$, then $D$ is biholomorphic to a disc possibly less a relatively closed polar set. In general dimension, they also proved a multi-dimensional Carathéodory theorem for such a biholomorphic, under the condition that the Bergman coordinate maps satisfy a certain regularity condition in terms of the Bergman kernel function. For precise statements, the reader is referred to their recent paper [DW2].

In this paper, we contribute three more results related to the curvature properties of Bergman metrics by using Calabi’s rigidity and extension theorems (Theorem 2.1 and Theorem 3.1). We first answer the aforementioned question by proving that the Bergman metric of a Stein manifold has a negative constant holomorphic sectional curvature if and only if it is biholomorphic to the ball with a certain pluripolar set being removed, relaxing the boundary blowing-up condition of Calabi diastasis functions in the work of Dong-Wong [DW1] . This result will be proved in $\S 5$ (Theorems 5.1, 5.2 and Corollary 5.3). In Theorem 6.1, we prove that a complex manifold, which may not be Stein, with its Bergman space being of infinite dimension (which is always the case when the manifold is a bounded domain in ${\mathbb{C}}^{n}$) can not have its Bergman metric positively constantly curved. Surprisingly, we find a domain in ${\mathbb{C}}^{2}$ whose Bergman metric has a constant holomorphic sectional curvature $2$. This domain has a three dimensional Bergman space and is not pseudoconvex. In Theorem 7.1, we prove that a complex manifold with a non-trivial bounded holomorphic function (which is certainly the case when the manifold is a bounded domain in ${\mathbb{C}}^{n}$) can not have a flat Bergman metric. We also construct an example which shows that a pseud-convex domain equipped with a well-defined Bergman metric may contain a codimension one totally geodesically embedded flat ${\mathbb{C}}^{n}$. However, it is not clear to us if there is a domain in ${{\mathbb{C}}}^{n}$ with a well-defined flat Bergman metric.

A main tool in this paper is Calabi’s work on extension and rigidity theorems for local isometric embeddings from a complex manifold equipped with a real analytic Kähler metric into the complex projective space of infinite dimension ${\mathbb{P}}^{\infty}$ [Ca]. In [HL], a simplified elementary proof of Calabi’s theorems, when the target space is of finite dimension, was given using ideas from CR geometry [BER]. In $\S 2$ and $\S 3$ of this paper, we continue the work in [HL] and present an elementary new proof of Calabi’s rigidity theorem ( Theorem 2.1) and Calabi’s extension theorem (Theorem 3.1) when the target is of infinite dimension. In $\S 4$, we provide an immediate application of Theorem 2.1 to prove Theorem 4.2. The last three sections contain the proofs of Theorems 5.1–5.2, Theorem 6.1 and Theorem 7.1, respectively.

Bergman kernels and metrics have been among the most extensively studied objects in complex analysis and geometry. To name a few, we refer the reader to related work in [Mo2], [EXX], [L1], [L2] as well as many references therein.

Acknowledgement: The authors would like to thank N. Mok and M. Xiao for many valuable conversations over the years related to the Calabi theory and Bergman metrics.

2 Holomorphic mappings into $\ell^{2}$ and ${\mathbb{P}}^{\infty}$

Let

[TABLE]

be the Hilbert space equipped with the standard $\ell^{2}$-norm defined above.

Let $M$ be a complex manifold of dimension $n$. In this paper, we always assume that $M$ is connected. A map $F=(f_{1},\cdots,f_{n},\cdots):M\to\ell^{2}$ is called a holomorphic map if each $f_{j}$ ($j\in{\mathbb{N}}$) is a holomorphic function in $M$ and if $\sum_{j=1}^{\infty}|f_{j}(z)|^{2}$ converges uniformly on any compact subset of $M$. Suppose that $M=U$ is a domain in ${\mathbb{C}}^{n}$. Then it follows that $\sum_{j=1}^{\infty}f_{j}(z)\overline{f_{j}(\overline{w})}$ converges uniformly on any compact subset of $U\times\hbox{Conj}(U)$ to a holomorphic function $H(z,w)$ in $U\times\hbox{Conj}(U)$. Thus $\sum_{j=1}^{\infty}\partial_{z}^{\alpha}f_{j}(z)\partial^{\beta}_{w}\overline{f_{j}(\overline{w})}$ converges uniformly on compact subsets of $U_{p}\times\hbox{Conj}(U)$ to $\partial_{z}^{\alpha}\partial_{w}^{\beta}H(z,w)$ for any multi-indices $\alpha,\beta\in({{\mathbb{Z}}}^{+})^{n}$. Here we write $\hbox{Conj}(U_{p}):=\{z:\ \overline{z}\in U\}$ and ${\mathbb{Z}}^{+}$ for the set of non-negative integers. In particular, for any $\alpha\in({{\mathbb{Z}}}^{+})^{n}$, $\partial_{z}^{\alpha}F(z)=(\partial_{z}^{\alpha}f_{1},\cdots,\partial_{z}^{\alpha}f_{m},\cdots)$ is a holomorphic map from $U$ into $\ell^{2}$. For any $z_{0}\in U$, as in the single-function case, one sees that $\sum_{|\alpha|\geq 0}\frac{\partial^{\alpha}_{z}F(z_{0})}{\alpha!}(z-z_{0})^{\alpha}$ converges uniformly in the $\ell^{2}$-norm to $F(z)$ for $z$ sufficiently close to $z_{0}$. For instance, suppose that $z_{0}=0$ and $D(0,1)=\{|z_{j}|\leq 1,\ j=1,\cdots,n\}\subset U$. Then $\|F(z)\|^{2}$ is bounded on $D(0,1)$. Write $F_{N}=\sum_{|\alpha|=0}^{N}\frac{\partial^{\alpha}_{z}F(0)}{\alpha!}z^{\alpha}$. Then by the Cauchy formula and thet Hölder inequality, we have for $|z|\leq\frac{1}{2}$ that

[TABLE]

Here $C_{1}$ and $C_{2}$ are constants independent of $z$ for $|z|\leq 1/2$. From this, it follows that $F_{N}$ converges uniformly to $F(z)$ for $|z|\leq\frac{1}{2}$ in the $\ell^{2}$-norm as $N\rightarrow\infty.$

Still let $F:M\rightarrow\ell^{2}$ be a holomorphic map. We denote by $X_{F}$ for the closure of the linear span of $F(M)$ in $\ell^{2}$. For any $p\in M$ and a small holomorphic coordinate $(U_{p},z)$ near $p$ with $z(p)=0$. Let $F=(f_{1},\cdots,f_{m},\cdots)$. Then $X_{F}$ is the closure $V^{o}$ of the linear span of vectors $\{\partial_{z}^{\alpha}F(z)|_{z=0}:\alpha\in({\mathbb{Z}}^{+})^{n}\}$. Apparently $\partial^{\alpha}F(z)\in X_{F}$. Thus, in particular, $V^{o}\subset X_{F}$. To see that $X_{F}\subset V^{o}$, let $\nu\in\ell^{2}$ be such that $\nu\perp V^{o}$. Then $h=<F,\nu>$ is a holomorphic function in $M$ which is identically zero near $p$. Hence $h\equiv 0$ and thus $X_{F}\subset V^{o}$. Hence, $X_{F}$ is the closure of the jets of $F$ at any given point in $M$. In particular, one sees that for any holomorphic function $h\not\equiv 0$ in $M$, $X_{hF}=X_{F}$.

Define ${\mathbb{P}}^{\infty}:=\ell^{2}\setminus\{0\}/\sim.$ Here $x\sim y$ for $x,y\in\ell^{2}\setminus\{0\}$ if and only if $x=ky$ for a certain $k\in{\mathbb{C}}\setminus\{0\}.$ For $x=(x_{1},\cdots,x_{n},\cdots)\in\ell^{2}\setminus\{0\}$, we write $[x_{1},\cdots,x_{n},\cdots]$ for its equivalence class in ${\mathbb{P}}^{\infty}$, called the homogeneous coordinates of the equivalence class of $x$. ${\mathbb{P}}^{\infty}$ is called the infinite dimensional complex projective space.

A map $[F]:M\rightarrow{\mathbb{P}}^{\infty}$ is called a holomorphic map from $M$ into ${\mathbb{P}}^{\infty}$ if there is a holomorphic representation of $[F]$ near each $p\in M$. Namely, for each $p\in M$, there is a small neighborhood $U_{p}$ of $p$ in $M$ such that $[F]=[f_{1},\cdots,f_{m},\cdots]$, where each $f_{j}$ with $j\in{\mathbb{N}}$ is a holomorphic function in $U_{p}$ and for each $q\in U_{p}$, there is a $k$, which may depend on $q$, such that $f_{k}(q)\not=0$. Moreover, $\sum_{j=1}^{\infty}|f_{j}(z)|^{2}$ is assumed to converge uniformly on compact subsets of $U_{p}$.

The generalized Fubini-Study metric, denoted by $\omega_{st}$, of ${\mathbb{P}}^{\infty}$ with homogeneous coordinates $[z_{1},\cdots,z_{n},\cdots]$ is formally defined by:

[TABLE]

Let $(M,\omega)$ be a Kähler manifold with a Kähler form $\omega$. Let $F=(f_{1},\cdots,f_{n},\cdots)$ be a holomorphic map from $M$ into $\ell^{2}\setminus\{0\}$. It then gives naturally a holomorphic map denoted by $[F]$ from $M$ into ${\mathbb{P}}^{\infty}$ such that

[TABLE]

Let $X\subset\ell^{2}$. $[X]=\{[x]:x\not=0,x\in X\}$ is called the projectization of $X$. Also for a set $E\subset{\mathbb{P}}^{\infty}$, we define $[E]_{c}=\{x\in\ell^{2}:[x]\in E\ \hbox{or }x=0\},$ called the cone associated with $E$. Let $E_{1},E_{2}$ be two non-zero closed subspaces of $\ell^{2}$. Suppose that $T$ is a linear isometric isomorphism from $E_{1}$ to ${E_{2}}$. $T$ induces a one to one and onto map $[T]$ from $[{E_{1}}]$ to $[{E_{2}}]$, which is called an isometric isomorphism from $[{E_{1}}]$ to $[{E_{2}}]$.

Let

[TABLE]

be a holomorphic map. If, for any a local holomorphic representation $\mathcal{F}=[f_{1},\cdots,f_{n},\cdots]$ over $U\subset M$, it holds that

[TABLE]

we call $\mathcal{F}$ a local holomorphic isometric embedding from $M$ into ${\mathbb{P}}^{\infty}$. Notice that we did not assume that $\mathcal{F}$ is one to one. Also, we define $X_{\mathcal{F}}=[X_{F}]$ for any local holomorphic representation $F$. By what we remarked above, $X_{\mathcal{F}}$ is independent of the choice of local holomorphic representations.

We now give an elementary proof of the following Calabi rigidity theorem through methods from CR geometry [DA] [HL]:

Theorem 2.1** (Calabi Rigidity Theorem).**

Let $M$ be a complex manifold. Let $\mathcal{F}$ and $\mathcal{G}:(M,\omega)\to{\mathbb{P}}^{\infty}$ be two holomorphic maps such that

[TABLE]

Then there is a linear isometric isomorphism $T$ from $X_{F}$ to $X_{G}$ such that ${\cal G}=[T]\circ{\mathcal{F}}$, where $F$ (or, $G$) is a local holomorphic representation of $\mathcal{F}$ (or $\cal G$, respectively).

Proof.

Let $(U,z)$ be a small holomorphic chart on $M$ with $z(U)$ a small ball centered at $0\in{\mathbb{C}}^{n}$. Assume that the holomorphic representations $F$ and $G$ of $\mathcal{F},\mathcal{G}$ on $U$, respectively, are given by

[TABLE]

with

[TABLE]

converging uniformly in $U$. Applying a linear isometric isomorphism (a unitary transformation) of $\ell^{2}$ and by shrinking $U$ if needed, we assume that $F_{1}=G_{1}\equiv 1$ and $F_{j}(0)=G_{j}(0)=0$ for $j\geq 2$. We then know that both $\partial_{z}^{\alpha}F(z)$ and $\partial_{z}^{\alpha}G(z)$ are holomorphic maps from $U$ into $\ell^{2}$ for any multiindex $\alpha$.

Notice that

[TABLE]

Then

[TABLE]

where $h$ is a certain holomorphic function in $U$ and $h(0)=0$. In the Taylor expansion of the left hand side at $z=0$, we only have mixed terms while the expansion of the right hand only has holomorphic or conjugate holomorphic terms. We thus conclude that $h\equiv 0$. Therefore,

[TABLE]

Moreover, through a complexification, one sees that

[TABLE]

By a lemma of D’Angelo [DA] (see also Calabi [Ca] or even in the book of Bochner-Martin [Section 5 of Chapter 2, BM] for related results), $F,G$ are different by an isometry as explained in detail below:

Let

[TABLE]

and set $X_{F}$ to be the closure of $X^{o}_{F}$ in $\ell^{2}$. Similarly we can define $X_{G}^{o}$ and $X_{G}$. Then $X_{F}$ and $X_{G}$ are the $\ell^{2}$-closure of $F(U),G(U)$, respectively.

Define $T:X_{F}\to X_{G}$ by letting

[TABLE]

and extend $T$ linearly to $X_{F}^{o}$. By (2.5), one has

[TABLE]

Hence

[TABLE]

Therefore $T$ extends to a well-defined linear isometry from $X^{o}_{F}\to X^{o}_{G}$ with inverse sending $\partial^{\alpha}G(0)$ to $\partial^{\alpha}F(0)$ for each $\alpha\in({\mathbb{Z}}^{+})^{n}$. The isometric property of $T$ follows from the following equality:

[TABLE]

[TABLE]

Thus $T$ can be extended to a linear isometric isomorphism from $X_{F}$ to $X_{G}$. For simplicity of notation, we still write $T$ for its extension to $X_{F}$. Then

[TABLE]

Define $[T]([x])=[Tx]$ for each $x\in[X_{\mathcal{F}}]$ with $x\not=0$. Then we proved that $[T]\circ[F(z)]=[T\circ F(z)]=[G(z)]$ over $U$. Notice that $[T]$ is a one to one and onto isometric transformation from $[X_{F}]$ to $[X_{G}]$. $[T]\circ[F(z)]=[T\circ F(z)]=[G(z)]$ over $M$ by applying the uniqueness theorem of holomorphic functions. ∎

3 Holomorphic extension along curves

We next give a simple and direct proof of a more complicated Calabi’s extension theorem:

Theorem 3.1** (Calabi Extension Theorem).**

Let $(M^{n},\omega)$ be an $n$-dimensional Kähler manifold with a real analytic Kähler form $\omega$. Let $U\subset M$ be a neighborhood of $z^{0}\in M$ and let $\mathcal{F}:U\to{\mathbb{P}}^{\infty}$ be a local holomorph isometric embedding. Then for any continuous curve $\gamma:[0,1]\rightarrow M$ with $\gamma(0)=z^{0}$, $\mathcal{F}$ extends holomorphically along $\gamma$ as a local isometric embedding.

Proof.

Let $\gamma:[0,1]\to M$ be a continuous map with $\gamma(0)=z^{0}$. We need to show that $\mathcal{F}$ extends holomorphically along $\gamma$ as a local holomorphic isometric embedding into ${\mathbb{P}}^{\infty}$. Suppose that there is $c\in(0,1)$ such that $\mathcal{F}$ extends holomorphically along $\gamma|_{[0,c)}$ as a local holomorphic isometric embedding, but it does not extend to a holomorphic isometric embedding along $\gamma|_{[0,c]}$ to a neighborhood of $\gamma(c)$.

We assume that $M$ near $\gamma(c)$ is a domain in ${\mathbb{C}}^{n}$ with coordinates $z$ and $z(\gamma(c))=0$. After a linear change of coordinates, we may assume that \omega|_{z=0}=i\partial\overline{\partial}\|z\|^{2}\big{|}_{z=0}. Hence, we can find a Káhler potential function $\psi(z,\overline{z})\geq 0$ with $\omega=i\partial\overline{\partial}\psi(z,\overline{z})$ near $z=0$ such that $\psi(z,z)=\|z\|^{2}+o(z\cdot\overline{z})$. Write

[TABLE]

Assume that $\phi(z,\xi)$ is holomorphic in $(z,\xi)$ for $\|z\|,\|\xi\|<4\delta_{1}<1$ for a certain $\delta_{1}>0$; and $\phi(z,\overline{z})>0,\ \phi(z,0)=1$ for $\|z\|<4\delta_{1}$ . In what follows, we write $D(0,\delta):=\{z\in{\mathbb{C}}^{n}:\ |z_{j}|<\delta,\ j=1,\cdots,n\}$ for $\delta>0$. Let $t_{0}<c$ be sufficiently close to $c$ such that $\gamma(c)\in D(\gamma(t_{0}),\frac{\delta_{1}^{2}}{3})$ and

[TABLE]

with $\phi_{0}$ holomorphic in $D(\gamma(t_{0}),3\delta_{1})$, $\phi_{0}(\gamma(t_{0}))=0$ and with $\Re{\phi_{0}}>-\frac{1}{2}$ in $D(\gamma(t_{0}),3\delta_{1})$.

After a simplification, we may write

[TABLE]

for $z\in D(\gamma(t_{0}),2\delta_{1})$. Here, $h_{0}=\log(1+\phi_{0})$ is holomorphic in $D(0,2\delta_{1})$ and $|b_{\alpha\overline{\beta}}|\leq C_{1}|\delta_{1}|^{-|\alpha|-|\beta|}$ for a certain constant $C_{1}$.

Since $\omega(z)=\mathcal{F}^{*}\omega_{st}$ near $\gamma(t_{0})$, we have

[TABLE]

where we assume without loss of generality that $\mathcal{F}=[1,f_{2},\cdots,f_{m},\cdots]$ near $z=\gamma(t_{0})$ with $\mathcal{F}(\gamma(t_{0}))=[1,0,\cdots,0,\cdots]$. Similar to an argument before, this shows that

[TABLE]

Hence, we obtain the following equation:

[TABLE]

For the rest of the proof, to simplify the notation, we assume $\gamma(t_{0})=0$. (This can be easily achieved through a translation). We thus have

[TABLE]

Note that

[TABLE]

with $C_{1}$ a certain constant independent of $\alpha$. Hence

[TABLE]

Thus

[TABLE]

converges uniformly on $D(0,{\delta_{2}})$ for any $0<\delta_{2}<\delta_{1}$. Thus each $f_{k}$ extends to a holomorphic function, denoted by $\widetilde{f_{k}}$, in $D(0,{\delta_{1}})$.

Next

[TABLE]

Here $C_{3}$ is a constant independent of $\alpha,k$. Hence $\sum_{k=2}^{\infty}|\widetilde{f_{k}}(z)|^{2}$ converges uniformly on compact subsets of $D(0,{\delta_{1}}^{2})$. Therefore $\mathcal{F}$ extends to a holomorphic map from $D(0,\delta_{1}^{2})$ into ${\mathbb{P}}^{\infty}$. By the uniqueness of real analytic functions, we have

[TABLE]

for $z\in D(0,\delta_{1}^{2})$. Since $\gamma(c)\in D(0,{\delta_{1}}^{2}/3)$, we obtain a contradiction. The proof of the theorem is complete. ∎

4 A uniformization theorem with a complete Bergman metric

Let $\Omega$ be a complex manifold of complex dimension $n\geq 1$. Write $\Lambda^{n}(\Omega)$ for the space of holomophic $(n,0)$-forms on $\Omega$ and define the Bergman space of $\Omega$ to be

[TABLE]

Then $A^{2}(\Omega)$ is a Hilbert space with an inner product defined below:

[TABLE]

Assume that $A^{2}(\Omega)\not=0$. Let $\{f_{j}\}_{1}^{N}$ be an orthonormal basis of $A^{2}(\Omega)$ with $N\leq\infty$ and define the Bergman kernel $(n,n)$-form [Ko] to be $K_{\Omega}=\sum_{j=1}^{N}f_{j}\wedge\overline{f}_{j}$. In a local holomorphic coordinate chart $(U,z)$ on $\Omega$, we have

[TABLE]

In this paper, we assume that $A^{2}(\Omega)$ is base-point free in the sense that $K_{\Omega}$ is nowhere zero on $\Omega$. We define a Hermitian $(1,1)$-form on $\Omega$ by $\omega_{\Omega}=\sqrt{-1}\partial\overline{\partial}\log K_{\Omega}(z,\overline{z}).$ We call $\omega_{\Omega}$ the Bergman metric on $\Omega$ if it induces a positive definite metric on $\Omega$.

When $\Omega\subset{\mathbb{C}}^{n}$ is a domain, we identify a holomorphic $n$-form $\omega=fdz_{1}\wedge\cdots\wedge dz_{n}$ with $f$. Thus $A^{2}(\Omega)$ reduces to the Hilbert space of $L^{2}$-integrable holomorphic functions, which is the classically defined Bergman space. Then we talk about the Bergman kernel functions which is a real analytic function over $\Omega$..

For an orthonormal basis $\{\phi_{j}\}_{j=1}^{N}$ of $A^{2}(\Omega)$, we define

[TABLE]

$\mathcal{F}$ is then well-defined, as we assumed that $A^{2}(\Omega)$ is base-point free, and is called a Bergman-Bochner map of $\Omega$.

$A^{2}(\Omega)$ is said to separate points in $\Omega$ if any Bergman-Bochner map of $\Omega$ is one-to-one. We say that $A^{2}(\Omega)$ separates holomorphic directions, if for any non-zero $X_{p}\in T_{p}^{(1,0)}$ there is a $\phi\in A^{2}(\Omega)$ such that $X_{p}(\phi)(p)\not=0$ in a local holomorphic chart near $p$ where $\phi$ is identified with a holomorphic function near $p$ as described above.

It is well-known [Ko] that when $A^{2}(\Omega)$ is base-point free and separates holomorphic directions at any point in $\Omega$, the Bergman metric of $\Omega$ is well-defined and becomes a biholomorphcially invariant metric.

Now $\sum_{j=1}^{N}\phi_{j}(z)\overline{\phi_{j}(z)}$ converges uniformly on compact subsets of $\Omega$ to a real analytic $(n,n)$-forms $K_{\Omega}(z,z)$ over $\Omega$, called the Bergman Kernel form of $\Omega$. Notice that for a different orthonormal basis $\{\psi_{j}\}_{j=1}^{N}$ of $A^{2}(\Omega)$, write $\psi_{j}=\sum_{k=1}^{N}a_{jk}\phi_{k}$. Then we have:

(${\bf 1}$): $\sum_{j}a_{jk}\overline{a_{j\ell}}=\delta_{k}^{\ell},\ \sum_{j}a_{kj}\overline{a_{\ell j}}=\delta_{k}^{\ell}.$

(${\bf 2}$): For $x,\ y\in{\mathbb{C}}^{N}$ or $\ell^{2}$ (when $N=\infty$), one has that $(\sum_{\ell=1}^{N}a_{\ell 1}x_{\ell},\cdots,\sum_{\ell=1}^{N}a_{\ell k}x_{\ell},\cdots)\in{\mathbb{C}}^{N}$ or $\ell^{2}$ ($N=\infty$) and

[TABLE]

Set $T:\ell^{2}\rightarrow\ell^{2}$ (when $N=\infty$) or $T:{\mathbb{C}}^{N}\rightarrow{\mathbb{C}}^{N}$ (when $N<\infty$) by

[TABLE]

Then $T$ is a linear isometric isomorphism. Moreover,

[TABLE]

Here $\mathcal{G}$ is a Bergman-Bochner map with $\mathcal{G}=[\psi_{1},\cdots,\psi_{k},\cdots]$ associated with the orthonormal basis $\{\psi_{j}\}$. The following is a simple but useful property for Bergman-Bochner maps.

Proposition 4.1**.**

Let $\Omega$ be a complex manifold with the base point free Bergman space $A^{2}(\Omega)$. Let $\mathcal{F}$ be the Bergman-Bochner map associated with the base $\{\phi_{j}\}_{j=1}^{N}.$ Then $X_{\mathcal{F}}={\mathbb{P}}^{N}$, where $0<N\leq\infty.$

Proof.

Assume without loss of generality that $N=\infty$. In a local holomorphic chart $(U,z)$, write $\phi_{j}=\phi_{j}^{0}dz_{1}\wedge\cdots\wedge dz_{n}$ for each $j$. We need only to prove $X_{F}=\ell^{2}$ where $F=(\phi^{0}_{1},\cdots,\phi^{0}_{m},\cdots)$. In fact, let $\nu=(\nu_{1},\cdots,\nu_{m},\cdots)\in\ell^{2}$ be such that $\nu\perp X_{F}$. Then $f(z)=\sum_{j=1}^{\infty}\overline{\nu_{j}}\phi_{j}$ is a holomorphic $n$-form that is identically zero over $U$ and thus over $\Omega$. Hence

[TABLE]

Thus $\nu=0$, or equivalently, $X_{F}=\ell^{2}.$ ∎

We define the following semi-positive Bergman form associated to $\Omega$ as follows:

[TABLE]

When $A^{2}(\Omega)$ further separates holomorphic directions in $\Omega$, it defines a Kähler metric called the Bergman metric as we mentioned earlier.

Then the Bergman-Bochner map $\mathcal{F}$ defined in (4.4) is a local holomorphic isometric embedding from $\Omega$ into ${\mathbb{P}}^{N}$ as

[TABLE]

Suppose that the holomorphic sectional curvature of $\omega_{\Omega}$ is a negative constant at each point and along each holomorphic direction of $\Omega$. Then by a result of Bochner [Bo], for any $p\in\Omega$ there is a biholomorphic map $f$ from a small neighborhood $U$ of $p\in\Omega$ into a certain domain in the unit ball ${\mathbb{B}}^{n}$ in ${\mathbb{C}}^{n}$ such that $f^{*}(\lambda\omega_{{\mathbb{B}}^{n}})=\omega_{\Omega}.$ Here $\lambda>0$ is a certain positive constant depending on the holomorphic sectional curvature of $(\Omega,\omega_{\Omega})$ and $\omega_{{\mathbb{B}}^{n}}$ is the Bergman metric of ${\mathbb{B}}^{n}$.

Notice that the Bergman metric of ${\mathbb{B}}^{n}$ is given by $\omega_{{\mathbb{B}}^{n}}=-(n+1)i\partial\overline{\partial}\log(1-\|w\|^{2})$, where $w=(w_{1},\cdots,w_{n})$ is the coordinates for $\mathbb{B}^{n}\subset\mathbb{C}^{n}.$ We use the Taylor expansion of $(1-x)^{-\mu}$ with $\mu=(n+1)\lambda$ at $x=0$ on $\{x\in\mathbb{R}:|x|<1\}$ to get

[TABLE]

The right hand side converges uniformly on compact subsets of $\mathbb{B}^{n}.$ This shows that for a certain sequence of monomials $\{P_{j}\}_{j=1}^{\infty}$ in $w$ we have $(1-\|w\|^{2})^{-\mu}=1+\sum_{j=1}^{\infty}|P_{j}(w)|^{2}$, which converges uniformly on compact subsets of $\mathbb{B}^{n}.$ This leads to a natural holomorphic isometric embedding $\mathcal{B}=[B]=[1,P_{1},\cdots,P_{j},\cdots]$ from $(X_{0},\mu\omega_{{\mathbb{B}}^{n}})$ to $\mathbb{P}^{\infty}.$ It is clear that $\mathcal{B}$ is one to one as each $w_{j}$ for $1\leq j\leq n$ (with an appropriate coefficient) is among the $P_{j}^{\prime}$s.

Now $\mathcal{B}\circ f:U\to{\mathbb{P}}^{\infty}$ is a holomorphic isometric embedding. By Calabi’s rigidity theorem, there is an isometric isomorphism $[T]:X_{\mathcal{F}}\rightarrow X_{\mathcal{B}}={\mathbb{P}}^{\infty}$ such that

[TABLE]

Here $\cal F$ is a Bergman-Bochner map of $\Omega$. When the dimension of $A^{2}(\Omega)$ is finite, we add zeros to the map to make it a map into ${\mathbb{P}}^{\infty}$. Notice that $[T]$ is induced by a linear isometry $T$ from $X_{F}$ to $X_{B}$. By the Calabi extension theorem, $\mathcal{B}\circ f$ extends along any curve in $\Omega$ initiated from $z_{0}\in U$ to a local holomorphic isometric embedding. Since each $w_{j}$ is a component in $\mathcal{B}$ up to a coefficient, we see that $f$ extends holomorphically and isometrically along any curve in $\Omega$ started at $z_{0}$. The extended map $\widetilde{f}$ apriori might be multi-valued. Since the normalized Bergman metric $\lambda\omega_{{\mathbb{B}}^{n}}$ is complete, $\widetilde{f}$ has image in ${\mathbb{B}}^{n}$. By the uniqueness of holomorphic functions, we have $[T]\circ\mathcal{F}(z)=\mathcal{B}\circ\widetilde{f}$ for any extended map $\widetilde{f}$ of $f$ through a curve starting at $z_{0}$. Since $\mathcal{B}$ is one to one, we see that $\widetilde{f}$ is independent of the choice of the chosen curves and thus gives a well defined local holomorphic isometry from $(\Omega,\omega_{\Omega})$ into $({\mathbb{B}}^{n},\lambda\omega_{{\mathbb{B}}^{n}})$. By the uniqueness of real analytic functions, $\widetilde{f}^{*}(\lambda\omega_{{\mathbb{B}}^{n}})=\omega_{\Omega}$ and thus $\widetilde{f}$ is a local biholomorphsm. In particular, $D=\widetilde{f}(\Omega)$ is a subdomain of ${\mathbb{B}}^{n}$. Moroever, when $A^{2}(\Omega)$ separates points and thus $\mathcal{F}$ is one to one, since $[T]$ is also one to one, $\widetilde{f}$ is then one to one. Notice that Bergman metrics are biholomorphic invariants. One has the following statement. (See also [HX1]):

Theorem 4.2**.**

Let $\Omega$ be complex manifold in ${\mathbb{C}}^{n}$. Assume its Bergman space is base point free. If its Bergman metric is well-defined and has a negative constant holomorphic sectional curvature on $\Omega$, then there is a surjective local biholomorphic map $\widetilde{f}:(\Omega,\omega_{\Omega})\to(D,\lambda\omega_{{\mathbb{B}}^{n}})$ where $D$ is a certain domain in ${\mathbb{B}}^{n}$ such that ${\widetilde{f}}^{*}(\lambda\omega_{{\mathbb{B}}^{n}})=\omega_{\Omega}$ for a certain positive constant $\lambda$. Moreover if $A^{2}(\Omega)$ separates points in $\Omega$, then $f$ is a biholomorphic map from $\Omega$ to $D$. In this case, the identity map $\hbox{Id}:(D,\omega_{D})\rightarrow(D,\lambda\omega_{{\mathbb{B}}^{n}})$ is an isometry, i.e., $\omega_{D}=\lambda\omega_{{\mathbb{B}}^{n}}|_{D}$ over $D$. Here $\omega_{D}$ is the Bergman metric of $D$.

When $\omega_{\Omega}$ is assumed to be complete, the inverse $g$ of $f$ extends holomorphically and isometrically along any path started at $w_{0}=f(z_{0})$ in ${\mathbb{B}}^{n}$ with image inside $\Omega$. (See, for instance, Proposition 11.4, [He]). Since ${\mathbb{B}}^{n}$ is simply connected, $g$ extends to a well-defined holomorphic isometry $\widetilde{g}$ from ${\mathbb{B}}^{n}$ into $\Omega$. Apparently $\widetilde{g}\circ\widetilde{f}=\hbox{Id}_{\Omega}$ and $\widetilde{f}\circ\widetilde{g}=\hbox{Id}_{{\mathbb{B}}^{n}}$. Thus $\widetilde{f}$ gives a biholomorphic isometry from $\Omega$ to ${\mathbb{B}}^{n}$. Since Bergman metrics are biholomorphic invariant, $(\Omega,\omega_{\Omega})$ is isometric to $({\mathbb{B}}^{n},\omega_{{\mathbb{B}}^{n}})$. Namely, one obtains the following uniformization theorem of Qi-Keng Lu:

Theorem 4.3** (Lu’s Uniformization Theorem).**

*Let $\Omega$ be a complex manifold. Assume its Bergman space is base point free and separates points. If its Bergman metric is a well-defined complete Kähler metric with a negative constant holomorphic sectional curvature, then there is a one-to-one and onto holomorphic isometric map $f:(\Omega,\omega_{\Omega})\to(B_{n},\omega_{B_{n}})$. *

5 A uniformization theorem on pseudo-convex domains

It is known that every domain $\Omega\subset{\mathbb{C}}^{n}$ with a complete Bergman metric is pseudocovex [Bre]. The converse is not true. A simple example is a domain of the form ${\mathbb{B}}_{n}\setminus V$ with $V$ a non-empty pure complex analytic variety of codimension one in ${\mathbb{B}}_{n}$. More generally, let $V\not=\emptyset$ be a pluripolar subset in $\Omega$. Namely, for a point $p\in E$, there is a neighborhood $U_{p}$ of $p$ and a non-constant plurisubharmonic $\rho_{p}$ over $U_{p}$ such that $U_{p}\cap E\subset\{z\in U_{p}:\ \rho_{p}(z)=-\infty\}$. Write $\Omega^{V}=\Omega\setminus V$. If the Bergman metric $\omega_{\Omega^{V}}$ has a negative constant holomorphic sectional curvature, then so is the Bergman metric $\omega_{\Omega}$. Thus when $\omega_{\Omega}$ is complete, there is a biholomorphic map $F$ from $\Omega$ to ${\mathbb{B}}^{n}$, whose restriction to $\Omega^{V}=\Omega\setminus V$ gives a biholomorphic isometry from $(\Omega^{V},\omega_{\Omega^{V}})$ to $({\mathbb{B}}_{n}\setminus V^{0},\omega_{{\mathbb{B}}_{n}\setminus V^{0}})$, where $V^{0}=F(V)$ is a pluripolar subset in ${\mathbb{B}}^{n}$. Notice that $(B_{n}\setminus V^{0},\omega_{{\mathbb{B}}_{n}\setminus V^{0}})$ is incomplete along $V^{0}$.

In this section, we prove an inverse version of the above statement. By Theorem 4.2, we need only to focus on subdomains in ${\mathbb{B}}^{n}$. As we mentioned in Introduction, under the blowing-up hypothesis of the Calabi diastasis function, the theorem is proved with a different method in the work of Dong-wong [DW1].

Theorem 5.1**.**

Let $D\subset{\mathbb{B}}^{n}$ be a pseudoconvex domain such that

[TABLE]

where $\omega_{D}$ is the Bergman metric of $D$. Then $E={\mathbb{B}}^{n}\setminus D$ is a pluripolar subset of ${\mathbb{B}}^{n}$. In particular, $\lambda=1$.

Proof.

Still write $\omega_{D}$ for the Bergman metric of $D$. Since Bergman metrics are biholomorphic invariant, applying an automorphism of ${\mathbb{B}}^{n}$, we assume that $0\in D$. Also, we let $\{\phi_{j}\}_{j=1}^{\infty}$ be an orthonormal basis of $A^{2}(D)$ with $\phi_{1}(0)>0$ and $\phi_{j}(0)=0$ for $j\geq 2$. Since $K_{D}(z,w)=\sum_{j=1}^{\infty}\phi_{j}(z)\overline{\phi_{j}(w)},$ we conclude that $K_{D}(z,0)={\phi_{1}(0)}\phi_{1}(z).$

Since $\omega_{D}=\lambda\omega_{{\mathbb{B}}^{n}}$ on $D$, we similarly have

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

For any element $h\in A^{2}(D)$, one has

[TABLE]

Here and in what follows, we write $dv(w)$ for the Lebeque measure of ${\mathbb{C}}^{n}$ with cordinates $w$. In particular,

[TABLE]

Therefore

[TABLE]

extends holomorphically to ${\mathbb{B}}^{n}$, as $K^{\lambda}_{{\mathbb{B}}^{n}}(z,w)$ is an anti-holomorphic function over $\overline{{\mathbb{B}}^{n}}$ in $w$, is holomorphic in $z\in{\mathbb{B}}_{n}$ and $K^{\lambda}_{{\mathbb{B}}^{n}}(z,w)$ is bounded in $w\in{\mathbb{B}}^{n}$ for any $z\in{\mathbb{B}}^{n}$.

Let $f_{0}(z)={1\over K_{D}(z,0)}$. Then $f_{0}(z)$ is holomorphic in ${\mathbb{B}}_{n}$ and $f_{0}(z)\not\equiv 0$. Let $Z(f_{0})=\{z\in{\mathbb{B}}^{n}:f_{0}(z)=0\}$. Then for any $h\in A^{2}(D)$ one has

[TABLE]

that extends to be a holomorphic function in ${\mathbb{B}}^{n}\setminus Z(f_{0})$.

Write $\partial D\cap{\mathbb{B}}^{n}=L_{1}\cup L_{2}$ with $L_{1}=\ \partial D\cap Z(f_{0})$ and $L_{2}=\partial D\cap{\mathbb{B}}^{n}\setminus Z(f_{0})$. For $z_{0}\in L_{2}$, by (5.4), one has

[TABLE]

Since $D$ is pseudoconvex, by a result of Pflug and Zwonek (Lemma 11 of [PZ]), for each$z_{0}$ in ${\mathbb{B}}^{n}\setminus L_{1}$ there is a neighborhood $U_{z_{0}}$ of $z_{0}$ such that $P_{z_{0}}=U_{z_{0}}\setminus D$ is a pluripolar set. Since $P_{z_{0}}$ has Hausdorff codimension at least two, $U_{z_{0}}\setminus P_{z_{0}}$ is connected. Thus $P_{z_{0}}=\partial D\cap U_{z_{0}}$ and $D\cap U_{z_{0}}=U_{z_{0}}\setminus P$. Moreover, $\overline{D}=\overline{{\mathbb{B}}^{n}}$. Now by the Josefson theorem [Jo], there is a plurisubharmonic function $\rho$ defined in ${\mathbb{C}}^{n}$ such that $L_{2}\subset\{\rho=-\infty\}$. Then $E:=\partial D\cap{\mathbb{B}}^{n}\subset\{\rho+\log|k_{D}(z,0)|=-\infty\}$. Namely, $\partial D\cap{\mathbb{B}}^{n}$ is a pluripolar set. Again since a (closed) pluripolar set has Hausdorff codimension at least two, ${\mathbb{B}}^{n}\setminus\partial D$ is connected which then has to be $D$ as $D$ is assumed to be connected. Namely, $D={\mathbb{B}}^{n}\setminus E$. Since a pluripolar set is a removable set for $L^{2}$-integrable holomorphic functions (Lemma 1, [Ir]) the Bergman metric of $D$ is just the restriction of $\omega_{{\mathbb{B}}^{n}}$ restricted to $D$. Hence $\lambda=1$.

This proves the theorem. ∎

Combining Theorem 4.2 with Theorem 5.1, we obtain the following theorem.

Theorem 5.2**.**

*Let $\Omega$ be a Stein manifold. Assume its Bergman space is base point free and separates points. Assume that its Bergman metric $\omega_{\Omega}$ is well-defined. Then $\omega_{\Omega}$ has a negative constant holomorphic sectional curvature if and only if there is a biholomorphic map $F:(\Omega,\omega_{\Omega})\to({\mathbb{B}}^{n}\setminus E,\omega_{{\mathbb{B}}^{n}}|_{{\mathbb{B}}^{n}\setminus E})$, where $E$ is a closed pluripolar subset of ${\mathbb{B}}^{n}$ such that ${\mathbb{B}}^{n}\setminus E$ is pseudoconvex. *

In particular, we have the following:

Corollary 5.3**.**

Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb{C}}^{n}$. Then its Bergman metric has a negative constant holomorphic sectional curvature if and only if $\Omega$ is biholomorphic to ${\mathbb{B}}^{n}\setminus E$ for a closed pluripolar subset of $E\subset{\mathbb{B}}^{n}$ that makes ${\mathbb{B}}^{n}\setminus E$ pseudoconvex.

6 Complex manifolds whose Bergman spaces are infinitely dimensional

In this section, we study the problem whether it is possible that the holomorphic sectional curvature of the Bergman metric of a complex manifold is a positive constant. Along these lines, we prove the following:

Theorem 6.1**.**

Let $\Omega$ be a complex manifold with its Bergman space of being infinite dimension. Suppose its Bergman space is base point free and separates points. Assume that its Bergman metric is well-defined. Then the Bergman metric of $\Omega$ does not have a positive constant holomorphic sectional curvature.

Proof.

Let $\Omega$ be a complex manifold with its Bergman metric being well defined and having a positive constant holomorphic sectional curvature. In a local holomorphic chart $(U,z)$ with $z(p)=0$ for a certain $p\in U$. Still write $U$ for $z(U)$ for simplicity of notation. Then there is a small neighborhood $U$ of $0\in\Omega$ and a biholomorphic map $F$ from $U$ to a neighborhood $V\subset{\mathbb{P}}^{n}$ of $[1,0,\cdots,0]$ such that $F^{*}(\omega_{st})=\lambda\omega_{\Omega}.$ Here $\omega_{st}$ is the Fubini-Study metric of ${\mathbb{P}}^{n}$. Then by the fact $(1+\|z\|^{2})^{\mu}$ can be written as a sum of squares of holomorphic functions if and only if $\mu$ is a natural number, we conclude that $\lambda=m$ is a nature number ([HX1]). Thus $({\mathbb{P}}^{n},m\omega_{st})$ can be isometrically embedded into ${\mathbb{P}}^{\infty}$ through a Veronese map. (See [HX1]). As in the negative constant case, by the Calabi rigidity and extension theorems, $F$ extends to a one-to-one and local biholomorphic map from $\Omega$ to its image $D\subset{\mathbb{P}}^{n}$. Let $X_{\infty}=\{[0,z_{1},\cdots,z_{n}]\in{\mathbb{P}}^{n}\}$. Replacing $\Omega$ by $\Omega\setminus F^{-1}(X_{\infty}\cap D)$, we may assume that $0\in D\subset{\mathbb{C}}^{n}$ which is identified as the cell with coordinates $\{[1,z_{1},\cdots,z_{n}]\}$. Hence $F$ is a biholomorphic isometry from $(\Omega,\omega_{\Omega})$ to $(D,m\omega_{st})$, where the Fubini-Study metric $\omega_{st}=i\partial\overline{\partial}\log(1+\sum_{j=1}^{n}|z_{j}|^{2}).$ Hence $\omega_{D}=m\omega_{st}$ as $F^{*}(\omega_{D})=\omega_{\Omega}$.

As before, let us assume that $\{\phi_{j}\}_{j=1}^{\infty}$ is an orthonormal basis of its Bergman space $A^{2}(D)$ with $\phi_{1}(0)>0$ and $\phi_{j}(0)=0$ for $j\geq 2$. Notice that $K_{D}(z,w)=\sum_{j=1}^{\infty}\phi_{j}(z)\overline{\phi_{j}(w)}$ and $K_{D}(z,0)={\phi_{1}(0)}\phi_{1}(z).$ Hence, we similarly have the following equation:

[TABLE]

Notice that there are finitely many holomorphic functions $\{\psi_{j}(z)\}_{j=0}^{N}$ such that (1): $\psi_{0}=1$ and $\psi_{j}=c_{j}z_{j}$ for $1\leq j\leq n$, where each $c_{j}$ is positive constant; and (2):

[TABLE]

Let

[TABLE]

Then $\Psi$ is also a holomorphic isometric embedding from $(D,m\omega_{st})$ into $({\mathbb{P}}^{\infty},\omega_{st}).$ By Theorem 2.1, we have a linear isometric isomorphism $T$ from $X_{H}$ to $X_{G}$ with

[TABLE]

such that

[TABLE]

Write $e_{j}=(0,\cdots,0,1,0,\cdots,)$ for the the element in $\ell^{2}$ with $1$ as its $j^{\rm{th}}$ componet and zero for all other componets. Denote by $\eta_{j}=T(e_{j})=(a_{1,j},\cdots,a_{k,j},\cdots)$ for $1\leq j\leq N$. Then there is a non-zero holomorphic function $h$ near [math] such that

[TABLE]

Since $\eta_{1},\cdots,\eta_{N+1},\eta_{N+k}$ are linearly dependent, there is a $(b_{1},\cdots,b_{N+1},x_{N+k})\not=0$ such that $\sum_{\ell=1}^{N+1}b_{\ell}\eta_{\ell}+b_{N+k}\eta_{N+k}=0$ for any $k\geq 2$. In particular, $\sum_{\ell=1}^{N+1}b_{\ell}a_{\ell,j}+x_{N+k}a_{N+k,j}=0$ for $j=1,\cdots,N+1$. Then $\sum_{\ell=1}^{N+1}b_{\ell}\phi_{\ell}+x_{N+k}\phi_{N+k}\equiv 0$. From this, we deduce that $\hbox{dim}_{{\mathbb{C}}}A^{2}(D)\leq N+1$, which is a contradiction. ∎

Domains with finite dimensional Bergman spaces were first constructed in [Wi]. See also recent work in [GGV]. We next present an example which demonstrates that the Bergman metric of a domain may have a positive constant holomorphic sectional curvature $2$ when its Bergman space is of finite dimension. Such a domain then has to be unbounded:

Let $\alpha>2$ and let

[TABLE]

Theorem 6.2**.**

For $2<\alpha<3$, the following statements hold:

(i) The Bergman space

[TABLE]

(ii) The Bergman metric has constant holomorphic sectional curvature $2$.

Proof.

We first notice that $D(\alpha)$ is a complete Reinhardt domain. We now compute that

[TABLE]

Therefore, if $\alpha\in(2,3)$, by symmetry in $z$ and $w$ variables, we have $\{1,z,w\}\in A^{2}(D)$. Since $D(\alpha)$ is Reinhardt, we have

[TABLE]

Next, we claim that $z^{p}w^{q}\not\in A^{2}(D)$ if $p+q\geq 2$ and $2<\alpha<3$. In fact,

[TABLE]

Therefore, since $D(\alpha)$ is a complete Reinhardt domain, $L^{2}$-integrable monomials form an orthogonal basis of $A^{2}(D(\alpha)$. Hence,

[TABLE]

Moreover $\|w\|_{A^{2}(D(\alpha))}=\|z\|_{A^{2}(D(\alpha))}$, by symmetry.

Now, the Bergman kernel function of $D(\alpha)$ can be written as

[TABLE]

where $c_{0},c_{1}$ are certain positive constant depending only on $\alpha$. Since it is isometric to the Fubini-Study metric of ${\mathbb{P}}^{2}$ restricted to a domain in the cell ${\mathbb{C}}^{2}$

[TABLE]

it has a constant holomorphic sectional curvature $2$.

∎

7 Complex manifolds with non-constant bounded holomorphic functions

We next prove that the Bergman metric of a complex manifold can not have zero holomorphic sectional curvature in general.

Theorem 7.1**.**

Let $\Omega$ be a complex manifold with a non-constant bounded holomorphic function. Suppose that its Bergman space $A^{2}(\Omega)$ is base point free and separates points in $\Omega$. Also assume that the Bergman metric of $\Omega$ is well-defined. Then the Bergman metric of $\Omega$ can not have constant zero holomorphic sectional curvature.

Proof.

Assume that $(\Omega,\omega_{\Omega})$ is as in the theorem. Let $(U,z)$ with $0\in z(U)$ be a holomorphic coordinate chart. (Still write $U$ for $z(U)$). Shrinking $U$ if needed, there is a biholomorphic map $F$ from $U$ to a neighborhood $V\subset{\mathbb{C}}^{n}$ of [math] such that $F^{*}(\omega_{eucl})=\omega_{\Omega},$ where $\omega_{eucl}=i\partial\overline{\partial}|z|^{2}$ is the Euclidean metric of ${\mathbb{C}}^{n}$. Then by the same application of the Calabi rigidity and extension theorems, we conclude that $F$ extends to a one-to-one and local biholomorphic map from $\Omega$ to its image $D\subset{\mathbb{C}}^{n}$ with $F^{*}(\omega_{eucl})=\omega_{\Omega}.$ Hence

[TABLE]

Still let $\{\phi_{j}\}_{j=1}^{\infty}$ be an orthonormal basis of the Bergman space $A^{2}(D)$ with $\phi_{1}(0)>0$ and $\phi_{j}(0)=0$ for $j\geq 2$. Then, as before, $K_{D}(z,w)=\sum_{j=1}^{\infty}\phi_{j}(z)\overline{\phi_{j}(w)}$ and $K_{D}(z,0)={\phi_{1}(0)}\phi_{1}(z).$ We thus have

[TABLE]

Therefore, by complexification, one has

[TABLE]

where

[TABLE]

Since $D$ is open and $0\in D$, there is a $r>0$ such that, the closed ball with radius $r$, $\overline{B(0,r)}\subset D$. Notice that

[TABLE]

Since

[TABLE]

from (7.3) and (7.4), it follows that

[TABLE]

This implies that $w^{\alpha}\phi_{1}(w)\in A^{2}(D)$ for each multi-indix $\alpha$. Therefore,

[TABLE]

In particular,

[TABLE]

Let $D_{k}\subset D,\ \ k=1,\cdots$ be an increasing sequence of smoothly bounded domains with $\bigcup_{k}D_{k}=D$ and write

[TABLE]

Since $\int_{D}|e^{\langle z,w\rangle}\overline{w}^{\beta}w^{\alpha}\phi_{1}(w)^{2}|dv(w)$ is a locally bounded function in $z\in D$ by (7.3) and (7.4), $J_{k}$ is holomorphic in $D$. Moreover, by (7.6)

[TABLE]

uniformly on compact subsets in $D$. Since ${\partial^{|\beta|}J_{k}(z)\over\partial z^{\beta}}=\int_{D_{k}}e^{\langle z,w\rangle}\overline{w}^{\beta}w^{\alpha}|\phi_{1}(w)|^{2}dv(w)$ which converges to $\int_{D}e^{\langle z,w\rangle}\overline{w}^{\beta}w^{\alpha}|\phi_{1}(w)|^{2}dv(w)$ uniformly for $z$ on compact subsets of $D$ as $k\to\infty$. Therefore,

[TABLE]

In particular, at $z=0$, one has

[TABLE]

This shows that $\{{z^{\alpha}\over\sqrt{\alpha!}}\phi_{1}\}_{|\alpha|=0}^{\infty}$ forms an orthonormal set for $A^{2}(D)$. Moreover,

[TABLE]

Therefore,

[TABLE]

Therefore, $\{{z^{\alpha}\over\sqrt{\alpha!}}\phi_{1}\}$ forms an orthonormal basis for $A^{2}(D)$. In particular, for any bounded holomorphic function $b(z)$ on $D$, one has $b(z)\phi_{1}\in A^{2}(D)$. Thus,

[TABLE]

and by Cauchy-Schwarz’s inequality,

[TABLE]

Here $\|b\|_{L^{\infty}(D)}=\sup_{z\in D}|b(z)|.$ Let

[TABLE]

which converges in ${\mathbb{C}}$. Then

[TABLE]

Hence $b(z)$ extends to an entire function on ${\mathbb{C}}^{n}$, denoted by $\widetilde{b}(z)$, and

[TABLE]

Since $D$ is biholomorphic to $\Omega$ that has non-constant bounded holomorphic functions, $D$ also possesses non-constant bounded holomorphic functions by pulling back those on $\Omega$.

Fix a non-constant bounded holomorphic function $b$ on $D$ that extends to an entire function $\widetilde{b}$. Then

[TABLE]

Since $\widetilde{b}$ is a non-constant entire function in ${\mathbb{C}}^{n}$, it is unbounded on ${\mathbb{C}}^{n}$. Pick a point $z^{0}\in{\mathbb{C}}^{n}$ such that

[TABLE]

For each $N\in{\mathbb{N}}$, we define

[TABLE]

Then

[TABLE]

By (7.13)–(7.17), one has

[TABLE]

This is a contradiction as $N$ can be arbitrarily large. The proof of Theorem 7.1 is complete. ∎

We are not able to find a domain, which has a flat Bergman metric. (This domain then must have no non-trivial bounded holomorphic functions by Theorem 7.1). However, we construct in what follows an unbounded pseudoconvex Hartogs domain $D\subset{\mathbb{C}}^{n+1}$ which contains a copy of $({\mathbb{C}}^{n},\omega_{eucl})$ as a totally geodesic complex submanifold of $(D,\omega_{D})$.

Let $D$ be the Reinhardt and Hartogs domain defined by

[TABLE]

Notice that $D$ is pseudo-convex as its defining function $\rho=2\log|w|+\|z\|^{2}$ is plurisubharmonic along its boundary. Also notice that $h(z,w)=w$ is a non-constant bounded holomorphic function over $D$. We will show in what follows that $({\mathbb{C}}^{n}\times\{0\},\omega_{eucl})$ is a total geodesy of $(D,\omega_{D})$. Namely, the holomorphic embedding $\iota:({\mathbb{C}}^{n},\omega_{eucl})\rightarrow(D,\omega_{D})$ with $\iota(z)=(z,0)$ is an isometric embedding. Moreover, the second fundamental form of ${\mathbb{C}}^{n}\times\{0\}$ in $D$ is identically zero, which is equivalent to the statement that holomorphic sectional curvatures of $(D,\omega_{D})$ are zero along ${\mathbb{C}}^{n}\times\{0\}$ by Gauss-Codazzi equation [Mo1].

We first prove that the Bergman space $A^{2}(D)$ is of infinite dimension with $\{z^{\alpha}w^{q}\}_{\alpha\geq 0,q\geq 0}$ forms an orthogonal basis of $A^{2}(D)$. To this aim, it suffices prove that

[TABLE]

as $D$ is a Reinhardt domain.

We next compute

[TABLE]

Therefore, the diagonal Bergman kernel function $K$ of $D$ is computed as follows:

[TABLE]

When $n=1$,

[TABLE]

and thus

[TABLE]

For a general $n$, when $w=0$, one has

[TABLE]

where $h(z,w)$ is a real analytic function in $D$. Write $z_{n+1}$ for $w$ in what follows and write the Bergman metric tension

[TABLE]

with $\alpha,\beta\in\{1,\cdots,n+1\}$. Clearly, $g_{(n+1)\overline{j}}=g_{j\overline{(n+1)}}=0$ for $1\leq j\leq n$. Also $\frac{\partial g_{i\overline{i}}}{\partial z_{k}}=\frac{\partial g_{i\overline{j}}}{{\partial\overline{z}_{l}}}=0$ when $w=0$ and $1\leq i,j,k\leq n$. Hence when $w=0$, $g^{j\overline{(n+1)}}=g^{(n+1)\overline{j}}=0$ for $j\leq n$. Now for $1\leq i,j,k,l\leq n$ and $w=0$, the Riemannian curvature tensions $R_{i\overline{j}k\overline{l}}$ are computed as follows [Mo1]:

[TABLE]

We thus conclude that when $w=0$, holomorphic sectional curvatures of the Bergman metric of $D$ along directions tangent to ${\mathbb{C}}^{n}\times\{0\}\subset D$ are zero.

By the Gauss-Godazzi equation for Kähler submanifolds [page 33, [Mo1]], we see that the second fundamental form of ${\mathbb{C}}^{n}\times\{0\}$ in $D$ is zero. Thus, ${\mathbb{C}}^{n}\times\{0\}$ with the flat metric is a totally geodesic submanifold in $(D,\omega_{D})$.

Closing Remark: Our study in this paper has raised several natural open questions. In Theorems 6.1 and 7.1, $\Omega$ might not be pseudoconvex. Instead, we assumed that $A^{2}(\Omega)$ is of infinite dimension in Theorem 6.1 and we assumed that $\Omega$ possesses a non-trivial bounded holomorphic functions in Theorem 7.1. A natural question is then to ask if these assumptions can be replaced by the Steinness of $\Omega$. More precisely, we make the following conjecture:

Conjecture 7.2**.**

Let $M$ be a Stein manifold of complex dimension at least two. Suppose that its Bergman space $A^{2}(M)$ is base-point free, separates points in $M$ and also separates holomorphic directions. Then the Bergman metric of $M$ cannot have a constant non-negative holomorphic sectional curvature.

There is an old folklore conjecture that asserts that the Bergman space of a Stein manifold is either trivial or of infinite dimension [JP] [Ju] [GGV]. If this folklore conjecture could be verified, by Theorem 6.1, an immediate consequence would be the affirmative answer to Conjecture 7.2 in the positive constant holomorphic sectional curvature case.