华罗庚域研究的综述

The Summarizations on Research of Hua Domain

华罗庚域的创建,统一了多复变中的对称典型域和蛋型域的研究,给多复变函数论提供了一个新的研究领域．对华罗庚域的研究,至今已经取得了一系列重要成果．本文简单介绍了华罗庚域创建的历史并着重介绍了华罗庚域上的Bergman核函数和Einstein-Khler度量的显表达式的计算,以及4个经典度量(Bergman度量,Carathéodory度量,Einstein-Kahler度量, Kobayashi度量)之间的等价关系,包括这些度量与Kobayashi度量的比较定理,阐述了Bergman度量等价于Einstein-Khler度量的这一丘成桐猜想在华罗庚域的特例Cartan-Hartogs域上也成立．着重指出了获得这些结果的新的思想和方法并提出了一些尚未解决的问题,以期更多的学者对华罗庚域感到兴趣并进行更深入的研究．

The Hua domain makes the Cartan domain and the egg domain into a unit, and provides a new research field for several complex variables. Up to now a series of important results were obtained. In this paper, the author introduces the brief history of Hua domain and introduces the computations for Berman kernel functions and Einstein-Kahler metrics on Hua domains. And discuss the equivalences between four classical metrics （Bergman, Carathéodory, Einstein-Kahler and Kobayashi metrics）. And prove that the Bergman metric is equivalent to the Einstein-Kahler metric on Cartan-Hartogs which is the special cases of Hua domains, that means the S-T Yau＇s conjecture is also true on Cartan-Hartogs domains. The author point out the new idea and new methods and give some open problems to expect the mathematicians interested in Hua domains and to lucubrate the Hua domains.