On the number of omitted values of entire quasiregular mappings
On the number of omitted values of entire quasiregular mappings
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关于整个拟正则映射的省略值个数
DOI：
10.1007/bf02797681
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
S. Rickman
中科院分区：
文献类型：
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作者：
S. Rickman
拟正则映射是平面上解析函数到实n维空间的自然推广。一个主要的公开问题是由Zori在[16]中首次提出的问题：对于n=>；3，一个非常数整拟正则映射f：R“~R”可以省略多少个值。这里猜想的最强形式是Picard定理的完全类似：F不能省略R“中的一个以上的点。文[4，4.4]和[10，定理2]证明了f不能省略一组正的共形容量。另一方面，f可以省略一点。对于n=3，在[16]中给出了这样一个例子，并且它可以以简单的方式推广到任何n。本文的目的是向Picard定理迈进一步。我们的结果是省略的值集始终是有限的，更准确地说，如下所示：
Quasiregular mappings form a natural generalization of the analytic functions in the plane to real n-dimensional space. One of the main open problems is the question, raised for the first time by Zori~ in [16], of how many values a nonconstant entire quasiregular mapping f: R"~ R" can omit for n=> 3. The strongest form of a conjecture here is the full analogue of Picard's theorem: f cannot omit more than one point in R". It was proved in [4, 4.4] and in [10, theorem 2] that f cannot omit a set of positive conformal capacity. On the other hand, f can omit one point. For n= 3 such an example was given in [16] and it can be generalized for any n in a straightforward manner. The purpose of this paper is to give a step towards Picard's theorem. Our result is that the set of omitted values is always finite, more precisely as follows: