Characterization of the unit ball in ℂn by its automorphism group
Characterization of the unit ball in ℂn by its automorphism group
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DOI：
10.1007/bf01403050
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发表时间：
1977-10
影响因子：
3.1
通讯作者：
B. Wong
中科院分区：
文献类型：
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作者：
B. Wong
备注。a）注意，条件ii）、iii）、iv）使i）成立的必要性是平凡的。此外，含义iii）~ ii）是显而易见的。因此，证明蕴涵式ii）=*，i）和iv）=~ ii）就足够了。B）在韦伯斯特的Berkeley论文（1975）和Burns和Shnider的一篇文章（[1]）中可以找到关于强伪凸超曲面的CR变换的一些密切相关的结果。此外，Burns在一封信中告诉作者，他和Shnider还利用Chern-Moser不变量和强伪凸域之间的双全纯映射的February Mans硬定理证明了这里给出的定理（在n= 2的情况下有一个稍微弱的版本）。然而，我们要指出的是，在我们的证明中，没有使用费曼定理和陈-莫泽不变量。我们将给出定理的两个证明。第一个是Diederich和Graham关于Bergman、Caratheodory和小林度规的边界行为的结果，以及对它们的全纯曲率的观察（[7]）。第二个是文[8]中导出的相应内禀测度的边界估计的应用。w 1.定义和已知结果
Remarks. a) Notice, that the necessity of conditions ii), iii), iv) for i) to hold is trivial. Furthermore, the implication iii)~ ii) is obvious. Therefore, it will suffice to prove the implications ii)=*, i) and iv)=~ ii). b) Some closely related results on CR transformations of strongly pseudoconvex hypersurfaces can be found in the Berkeley thesis (1975) of Webster and an article of Burns and Shnider ([1]). Furthermore, Burns told the author in a letter that he and Shnider also proved the theorem given here (with a slightly weaker version in the case n= 2) by using Chern-Moser invariants and Feffermans hard theorem on biholomorphic mappings between strongly pseudoconvex domains. However, we want to point out, that in our proof Feffermans theorem and also Chern-Moser invariants are not used. We shall give two proofs of our theorem. The first one involves the results of Diederich and Graham on the boundary behavior of the Bergman, Caratheodory and Kobayashi metrics, together with an observation on their holomorphic curvatures ([7]). The second one is an application of the boundary estimates for the corresponding intrinsic measures which were derived in [8]. w 1. Definitions and Known Results