The explicit forms and zeros of the Bergman kernel function for Hartogs type domains

We define the Cartan–Hartogs domain, which is the Hartogs type domain constructed over the product of bounded Hermitian symmetric domains and compute the explicit form of the Bergman kernel for the Cartan–Hartogs domain using the virtual Bergman kernel. As the main contribution of this paper, we show that the main part of the explicit form of the Bergman kernel is a polynomial whose coefficients are combinations of Stirling numbers of the second kind. Using this observation, as an application, we give an algorithmic procedure to determine the condition that their Bergman kernel functions have zeros.     

第四类超Cartan域的Bergman核函数

显式给出了第四类超Cartan域的Bergman核函数及其全纯自同构群．     

The Bergman kernels on super-Car tan domains of the first type

The Bergman kernel function and biholomorphic automorphism group for super-Cartan domain of the first type are given in explicit formulas.     

Bergman kernel on generalized exceptional hua domain

We have computed the Bergman kernel functions explicitly for two types of generalized exceptional Hua domains, and also studied the asymptotic behavior of the Bergman kernel function of exceptional Hua domain near boundary points, based on Appell's multivariable hypergeometric function.     

有界域的Bergman核函数显式表示的最新进展

对多维复数空间的有界域，如何求出它的Bergman核函数的显表达式，是多复变研究中的一个重要方向。本文综述了迄今为止的所有重要结果以及方法上的进展，特别对新近引进的华罗域，综述了它们的Bergman核函数的显表达式及其计算方法上的创新。     

The Bergman kernel of the symmetrized polydisc in higher dimensions has zeros

We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros. Received: 15 November 2005; revised: 10 January 2006     

Effective methods of determining the modulus of doubly connected domains

The modulus of a doubly connected domain is determined by a quotient of certain kernel functions, namely the Bergman kernel, the reduced Bergman kernel and the square of the Szegö kernel. These methods are more efficient than methods involving the curvature of the Bergman metric.     

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.     

The Bergman Kernel Method for the Numerical Conformal Mapping of Simply Connected Domains

A numerical method for the conformal mapping of simply-connecteddomains onto the unit disc is considered. The method is basedon the use of the Bergman kernel function of the domain. Itis shown that, for a successful application, the basis of theseries representation of the kernel must include terms thatreflect the main singular behaviour of the kernel in the complementof the domain.     

Lu Qi-Keng conjectue and Hua domain

The first part of this paper discusses the motivation for the Lu Qi-Keng conjecture and the results about the presence or the absence of zeroes of the Bergman kernel function of a bounded domain in C~n.Its second part summarizes the main results on the Hua domains,such as the explicit Bergman kernel function,the comparison theorem for the invariant metrics,the explicit complete Einstein-K(?)hler metrics,the equivalence between the Einstein-K(?)hler metric and the Bergman metric,etc.     

Bergman spaces of higher‐dimensional hyperbolic polyhedron‐type domains I

In this paper we develop explicit formulas for the Green's function and the monogenic reproducing Bergman kernel function of some hyperbolic polyhedron‐type domains that generalize the fundamental domain of the modular group SL(2,?) to higher dimensions. Copyright © 2005 John Wiley & Sons, Ltd.     

实解析变换下Bergman核函数变换公式及其应用

利用形变理论研究实解析变换下(加权)Bergm an核函数变换公式,并利用这一公式从已知域的Bergm an核函数求得新的域的加权Bergm an核函数.我们的结果推广了经典的在双全纯映照下的Bergm an核函数变换公式.     

On the Navier–Stokes equations with free convection in three‐dimensional unbounded triangular channels

The quaternionic calculus is a powerful tool for treating the Navier–Stokes equations very elegantly and in a compact form, through the evaluation of two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. In this paper, we use special variants of quaternionic‐holomorphic multiperiodic functions in order to obtain explicit formulas for unbounded three‐dimensional parallel plate channels, rectangular block domains and regular triangular channels. Copyright © 2007 John Wiley & Sons, Ltd.     

A Domain Integral Equation for the Bergman Kernel

Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.^{1 2}     

第一类Cartan—Egg或的Bergman核函数

以显式给出了第一类Cartan-Egg域的Bergman核函数及其全纯自同构群。     

加权调和Bergman核及其诱导的度量矩阵

在Rn中的有界域上建立加权调和Bergman核,并得出单位球的加权调和Bergman核的表达式;利用加权调和Bergman核在Rn的有界域上构造度量矩阵;得到关于调和映射的Jacobi矩阵与度量矩阵之间的一个不等式.     

第三类华罗庚域的Bergman核函数

本文主要是计算第三类华罗庚域的Bergman核函数的显式表达式.由于华罗庚域既不是齐性域又不是Reinhardt域,故以往求Bergman核函数的方法都行不通.本文用新的方法进行计算.关键之处有两点:一是给出第三类华罗庚域的全纯自同构群,群中每一元素将形为(W,Z0)的内点映为点(W*,0);二是引进了semi—Reinhardt的概念并求出了其完备标准正交函数系.     

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂ ⁿ satisfying the condition R. Suppose that its Bergman kernel extends to [`(W)]×[`(W)]\overline{\Omega}\times\overline{\Omega} minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.     

Bergman kernel function on the third Hua Construction

The Bergman kernel function for Hua Construction of the third type is given in an explicit formula.     

The Bergman Kernel Function and Full Group of Holomorphic Automorphism on a Reinhardt Domain

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