有限连通平面区域的全纯映射
Holomorphic mappings of finitely connected planar domains
AI总结：
本文探究无需Koebe定理证明有限连通平面区域全纯映射相关结果，给出连通度>2的有限连通区域间真全纯映射的刚性定理，还提供不依赖欧拉示性数的平面Riemann-Hurwitz公式新初等证明。
AI中文摘要：
关于有限连通平面区域全纯映射的结果常利用Koebe著名的圆映射定理证明，典型例子是Julia对连通度k≥3的有限连通平面区域自同构群大小的界，以及Heins后续得到的精确界。本文旨在探究是否能在不使用Koebe定理的情况下证明这些结果及其他相关结果。主要结果是关于连通度大于2的有限连通区域之间真全纯映射的刚性定理，该定理可立即根据两个此类区域的连通度给出它们之间真全纯映射数量的界。利用相同思路，本文还提供了平面Riemann-Hurwitz公式的新初等证明，该证明不使用欧拉示性数。我们的大部分证明仅依赖Riemann映射定理、Schwarz反射原理和辐角原理。
英文摘要：
Results on holomorphic mappings of finitely connected planar domains are often proved using Koebe's famous circle mapping theorem. A prominent example is Julia's bound on the size of the automorphism group of finitely connected planar domains of connectivity $k\geq3$ and the sharp bound obtained later by Heins. The purpose of this article is to explore whether it is possible to give a proof of these results, and several other related results, without using Koebe's theorem. The main result is a rigidity theorem for proper holomorphic maps between finitely connected domains of connectivity higher than $2$ that immediately yields a bound on the number of proper holomorphic mappings between two such domains in terms of their connectivities. Using the same ideas, we also provide a new elementary proof of the planar Riemann--Hurwitz formula that does not use the Euler characteristic. Most of our proofs rely only on the Riemann mapping theorem, the Schwarz reflection principle, and the argument principle.
