The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature , which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.     

The Density Problem for Unbounded Bergman Operators

The unbounded Bergman operator, the operator of multiplication by on an unbounded open subset of the plane, is considered. We give a complete answer regarding the density problem of unbounded Bergman operators in terms of its equivalence to the problem of bounded point evaluations for the Bergman spaces. Using this equivalence and the notion of Wiener capacity, we obtain simple geometric conditions that classify almost those open subsets of the plane for which the corresponding Bergman operators are densely defined. With the aid of an analytic approach, we are also able to give condition for a large collection of open subsets of the plane for which all the positive integer powers of the corresponding Bergman operators are densely defined. Submitted: December 14, 2001? Revised: January 14, 2001.     

Intrinsic derivative,curvature estimates and squeezing function

This survey paper consists of two folds. First of all, we recall the concept of intrinsic derivative which was introduced by Lu(1979) and the related works due to Lu in his last ten years, including the holomorphically isometric embedding into the infinite dimensional Grassmann manifold and the Bergman curvature estimates for bounded domains in C~n. Inspired by Lu's idea, we give the lower and upper bounds estimates for the Bergman curvatures in terms of the squeezing function—one concept originally introduced by Deng et al.(2012).Finally, we survey some recent progress on the asymptotic behaviors for Bergman curvatures near the strictly pseudoconvex boundary points and present some open problems on the squeezing functions of bounded domains in C~n.     

UNIFORM ESTIMATES OF SOLUTIONS OF (δ)-EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for (p, q) differential forms on an open set with C1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of (δ)-equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of (δ)-equation for (p, q) differential forms on a Stein manifold.     

The Bergman metric on complete Kähler manifolds

We use the existence of a bounded uniformly Hölder continuous plurisubharmonic exhaustion function to characterize the Bergman completeness of a complete Kähler manifold. As an application, we proved that any simply-connected complete Kähler manifold with sectional curvature bounded above by a negative constant is Bergman complete. Mathematics Subject Classification (2000):32H10Supported by NSFC grant no. 10271089     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

复Finsler流形上的Koppelman-Leray-Norguet公式

利用不变积分核(Berndtsson核),复Finsler度量和联系于Chern-Finsler联络的非线性联络,研究复Finsler流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的积分表示,得到了(p,q)型微分形式的Koppelman-Leray-Norguet公式和■-方程的解．作为应用,利用复Finsler度量和联系于Chern-Finsler联络的非线性联络,给出了Stein流形上具有逐块光滑C~((1))边界的有界域上(p,q)型微分形式的Koppelman- Leray-Norguet公式以及■-方程的解,并且得到了Stein流形上实非退化强拟凸多面体上(p,q)型微分形式的积分表示式和■-方程的解．     

Bergman空间的实变刻画

介绍复球上Bergman空间实变理论的某些新进展,包括Bergman空间关于Carleson矩形的实变原子分解,极大函数和面积函数刻画以及运用帐篷空间的实变刻画.特别是,用齐次空间上向量值Calderón-Zygmund奇异积分算子理论研究Bergman积分算子的Lp有界性并给出了Bergman空间面积积分刻画的新证明...     

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ?×?^?, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ?×?^? of recent results of Wold and Forstneri? on the long-standing problem of properly embedding open Riemann surfaces into ?², with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.     

On complex Banach manifolds similar to Stein manifolds

We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.     

A Remark on Steinness

In this paper, the authors prove that if Mn is a complete noncompact Kahler manifold with a pole p, and its holomorphic bisectional curvature is asymptotically non-negative to p, then it is a Stein manifold.     

A Simple Proof of the Riemann Mapping Theorem for Domains with Spherical Boundary

We give a simple proof of a Riemann mapping theorem for domains in a Stein manifold with a spherical boundary. Those boundaries are real hypersurfaces which are locally CR equivalent to the sphere inside complex space. The main observation is that for higher dimensional Stein manifolds, the fundamental group of its boundary coincides with that of its interior.     

The Bergman kernel functions on Hua domains

We get the Bergman kernel functions in explicit formulas on four types of Hua domain.There are two key steps: First, we give the holomorphic automorphism groups of four types of Hua domain; second, we introduce the concept of semi-Reinhardt domain and give their complete orthonormal systems. Based on these two aspects we obtain the Bergman kernel function in explicit formulas on Hua domains.     

Wu's theorem for Kähler–Finsler spaces

In this paper, we study strongly convex Kähler–Finsler manifolds. We prove two theorems: A strongly convex Kähler–Berwald manifold with a pole is a Stein manifold if it has nonpositive horizontal radial flag curvature; A strongly convex Kähler–Finsler manifold with a complex pole is a Stein manifold if it has nonpositive horizontal radial flag curvature.     

A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert-Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.     

On the Bergman Theory for Solenoidal and Irrotational Vector Fields. II. Conformal Covariance and Invariance of the Main Objects

This is a continuation of our work (González-Cervantes et al. in On the Bergman theory for solenoidal and irrotational vector fields. I. General theory. Operator theory: advances and applications. Birkhauser, accepted) where for solenoidal and irrotational vector fields theory as well as for the Moisil–Théodoresco quaternionic analysis we introduced the notions of the Bergman space and the Bergman reproducing kernel and studied their main properties. In particular, we described the behavior of the Bergman theory for a given domain whenever the domain is transformed by a conformal map. The formulas obtained hint that the corresponding objects (spaces, operators, etc.) can be characterized as conformally covariant or invariant, and in the present paper we construct a series of categories and functors which allow us to give such characterizations in precise terms.     

Smoothly parameterized Čech cohomology of complex manifolds

A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Čech theory. If however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.     

Holomorphic Campanato type spaces over Carleson tubes and Bergman metric balls

In this paper we give some characterizations of holomorphic Campanato type spaces in terms of Carleson tubes and Bergman metric balls in the unit ball.     

Characterizations of the harmonic Bergman space on the ball

In the harmonic Bergman space with the normal weight, we prove norm equivalences in terms of radial, gradient and invariant gradient norms. Using this, we give new characterizations in terms of Lipschitz type conditions with Euclidean, pseudo-hyperbolic and hyperbolic metrics on the ball.