Proper holomorphic mappings of balanced domains in $$\mathbb {C}^n$$ C n

Abstract

Abstract. We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in C n , n > 1. Specifically: we prove that any proper holomorphic selfmap of a certain type of balanced, finite-type domain in C n , n > 1, is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein's theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result.

Proper slice regular functions over quaternions

“…Alexander 定理得到了广泛关注, 被推广到各种区域上, 如强拟凸域 [10] 、具有实解析边界的拟凸 域 [11,12] 、推广的椭球体 [13] 、秩大于 2 的不可约有界对称域 [14][15][16] 、有界圆型域 [17] 、平衡域 [18,19] 和 华域 [20] . 关于全纯逆紧映射的更多研究成果可以参见早期的综述文献 [21,22].…”