K(a)hler-Einstein度量和Bergman度量的等价问题

华罗庚域的特殊类型Cartan-Hartogs域YⅡ(N,p;K)当K=p/2+1/p+1时,求解了该域上的复Monge-Ampère方程的边值问题,从而得到该域的完备K(a)hler-Einstein度量的显表达式,并且得到此度量下的全纯截曲率的负的上下确界,最后证明了此K(a)hler-Einstein度量与Bergman度量等价.     

第三类超Cartan域上的比较定理

本文给出了第三类超Cartan域上不变Kalher度量下的全纯截曲率的表达式.利用其Bergman度量的完备性,构造了一个不比Bergman度量小的完备的不变Kalher度量,证明了在此Kalher度量下的全纯截曲率有一个负上界,从而证明了第三类超Cartan域的Bergman度量与Kobayashi度量的比较定理.     

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.     

A Characterization of Complete Bounded Domain

1 IntroductionThis paper is concerned with biholomorphic mappings between two bounded domains D andG both in C". Consequently, an important question is whether the domain D is biholomorphicto G? We give an answer for this question under a very weak condition.Let D be a bounded domain and Bn the unit ball in Cn. Let T(D) be the holomorphictangent bundle of D. We will identify T(D) with D × Cn. Let H(D1, D2) be the family ofholomorphic mappings from D1 to D2. We introduce the followin…     

The Bergman Kernels and Comparison Theorem on Hua Domain of the Third Type

1 The Bergman Kernel Function The Bergman Kernel function plays an important role in several complexvariables. L.～K. Hua［1］ obtained the Bergmankernel, Cauchy kernel and Poisson Kernel on four types of Cartan domains.So the basic theory of several complex variables was built on such domains.Yin Weiping［2］ computed such three types of kernels on two exceptional Cartandomains. The Cartan domains are bounded symmetric domain (homogeneous domainalso). For the bounded nonsymmetric domainwe can get the explicit formulas of Bergman kernel function by using theL.～K. Hua's method［3,4］ if the holomorphic transitivegroup is given. Except the bounded homogeneous domains, we have known theexplicit formulas of Bergman kernelfunction on egg domain E(p1, p2,,pn) defined by the inequality |z1|2p1)++|zn|2)/(pn)< 1 in some cases. For example, Bergman［5］computed the Bergman kernel of E(1,p) in C2.D'Angelo used a method to sum an infinite series to get the explicit formulaof Bergman kernel forE(1,2,,1, pn). Where the z C［6］ or z Cm［7］. When p1, p2,, pn are positive integers,the explicit formula of the Bergman for E(p1,p2,,pn) was first computedby Zinov'ev［8］. If p1,p2,,pn-1 are positive integers and pn is apositive real number. The explicit formula of Bergman kernel ofE(p1,, pn) is given by Francsics and Hanges［9］. Recently, H.～Boas,Siqi Fu and J. Straube［10］ proved two principles to compute the Bergman kernelsin a simple way.     

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。     

Khler-Einstein度量和Bergman度量的等价问题

华罗庚域的特殊类型Cartan-Hartogs域Y_Ⅱ(N,p;K)当K=p/2 1/(p 1)时,求解了该域上的复Monge-Ampère方程的边值问题,从而得到该域的完备K■hler-Einstein度量的显表达式,并且得到此度量下的全纯截曲率的负的上下确界,最后证明了此K■hler-Einstein度量与Bergman度量等价。     

The Comparison Theorem on Cartan-Hartogs Domain of the Fourth Type

As is known to all, theory of invariant metric is very important in several complex analysis. The Bergman, Caratheodory and Kobayashi metrics are important biholomorphic invariants. They play very important role in studying the boundary geometry of the domain and biholomorphic mappings extending smoothly to the boundaries of the relevant domains.     

第四类华罗庚域的Bergman核函数

本文给出了第四类华罗庚域H EIV上的Bergman核函数,全纯自同构群以 及semi—Reinhardt域上的完备标准正交函数系.