Holomorphic mappings from the ball and polydisc
Holomorphic mappings from the ball and polydisc
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DOI：
10.1007/bf01351851
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发表时间：
1974-09
影响因子：
1.4
通讯作者：
Herbert J. Alexander;Herbert J. Alexander
中科院分区：
文献类型：
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作者：
Herbert J. Alexander;Herbert J. Alexander
介绍。在~ Ltm中，开单位球B的全纯自同胚（“自同构”）是由在邻域Bn上全纯的某些有理函数给出的，并推导出球的边界bB的同胚。我们的第一个结果可以看作是这些自同构的一个局部表征：对于n> 1，一个定义在bB点的邻域内的非常全纯映射到~”，并且映射到bB自身的非常全纯映射必然是一个自同构，或者更准确地说，扩展为一个自同构。应用这一方法，我们得到了关于Bn的每一个固有全纯自映射是否为自同构这一尚未解决的问题的一些信息。特别是，我们恢复了Pelles（[3,5]）的结果（Cor. 1.1）。在第二部分中，我们考虑了多盘的全纯映射。根据庞加莱6的经典定理，c2中的多盘U z与球B2不存在生物全纯性。最后，我想承认，上述自同构的特征可能已经为已故的L6wner教授所知，至少对于两个复杂变量。我要感谢L. Bers教授和C. Titus教授提供了他们与L6wner口头交流的信息。
Introduction. The holomorphic self-homeomorphisms (" automorphisms") of the open unit ball B, in~ Ltm have long been known [1]-they are given by certain rational functions which are holomorphic on a neighborhood of Bn and induce a homeomorphism of the boundary, bB,, of the ball. Our first result can be viewed as a local characterization of these automorphisms: For n> 1, a nonconstant holomorphic mapping into~" which is defined in a neighborhood of a point of bB, and which maps bB, into itself is necessarily an automorphism, or, more precisely, extends to be an automorphism. We apply this to obtain some information on the as yet unsettled question as to whether every proper holomorphic self-mapping of Bn is an automorphism. In particular, we recover (Cor. 1.1) a result of Pelles ([3, 5]). In the second part, we consider holomorphic mappings from polydiscs. According to a classical theorem of Poincar6, there exists no biholomorphism from the polydisc U z in C 2 with the ball B2. We obtain some integral formulas which yield a quantitative explanation of this phenomenon, Finally I wish to acknowledge that the above characterization of automorphisms may have been known to the late Professor L6wner, at least for two complex variables. I want to thank Professors L. Bers and C. Titus for this information on their oral communication with L6wner.