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。     

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.     

New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym-metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux-iliary tool, we compute the integral of complex powers of the generic norm on a boundedsymmetric domains using the well-known integral of Selberg. This generalizes matrix in-tegrals of Hua and leads to a special polynomial with integer or half-integer coefficientsattached to each irreducible bounded symmetric domain.     

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.     

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.     

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.     

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.     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1,…,1/(pr-1) are positive integers and 1/pr is an arbitrary positive real number.     

Bergman kernel function on Hua Construction of the second type

In this paper, we give an explicit formula of the Bergman kernel function on Hua Construction of the second type when the parameters 1/p1, ..., 1/p_r−1 are positive integers and 1/p_r is an arbitrary positive real number.

     

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.     

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．     

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。     

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.     

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.     

A note on the Bergman kernel of a certain Hartogs domain

We give an explicit formula of the Bergman kernel of a certain Hartogs domain.     

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.

     

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.     

Asymptotics of orbifold Bergman kernels: mixed curvature case

We derive some asymptotic expansion formulas of the Bergman kernel of high tensor powers of an Hermitian orbifold line bundle with mixed curvature tensored with an orbifold vector bundle on a compact symplectic orbifold. In particular, when the orbifold has isolated singularities, we get an explicit formula for the asymptotic expansion of the Bergman kernel in the distribution sense. Finally, by applying our results to the complex case, we get a Riemann–Roch–Kawasaki type formula.     

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.     

第四类Hua结构的Bergman核函数

本文给出了第四类Hua结构的Bergman核函数及其全纯自同构群．