# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Proper holomorphic mappings between invariant domains in $\mathbb {C}^n$HTML articles powered by AMS MathViewer

by Jiafu Ning, Huiping Zhang and Xiangyu Zhou
Trans. Amer. Math. Soc. 369 (2017), 517-536 Request permission

## Abstract:

In the present paper, we prove the following result generalizing some well-known related results about biholomorphic or proper holomorphic mappings between some special domains in $\mathbb {C}^n$. Let $G_1$ and $G_2$ be two compact Lie groups, which act linearly on $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_j}=\mathbb {C}$ for $j=1,2$. Let $0\in \Omega _j$ be bounded $G_j$-invariant domains in $\mathbb {C}^n$ for $j=1,2$. If $f:\Omega _1\rightarrow \Omega _2$ is a proper holomorphic mapping, then $f$ extends holomorphically to an open neighborhood of $\overline {\Omega }_1$, and in addition if $f^{-1}(0)=\{0\}$, then $f$ is a polynomial mapping. We also prove that if $0\in \Omega$ is a $G_1$-invariant pseudoconvex domain in $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_1}=\mathbb {C}$, then $\Omega$ is orbit convex. The second result is used to prove the first one.
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• Jiafu Ning
• Affiliation: College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People’s Republic of China
• Email: jfning@cqu.edu.cn
• Huiping Zhang
• Affiliation: Department of Mathematics, Information School, Renmin University of China, Beijing 100872, People’s Republic of China
• MR Author ID: 717149
• Email: huipingzhang@ruc.edu.cn
• Xiangyu Zhou
• Affiliation: Institute of Mathematics, Academy of Mathematics and Systems Science, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
• MR Author ID: 260186
• Email: xyzhou@math.ac.cn
• Received by editor(s): December 18, 2013
• Received by editor(s) in revised form: August 19, 2014, and January 9, 2015
• Published electronically: May 6, 2016
• Additional Notes: The authors were partially supported by NSFC. The first author was supported by the Fundamental Research Funds for the Central Universities (Project No.0208005202035)
The second author is the corresponding author
• © Copyright 2016 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 369 (2017), 517-536
• MSC (2010): Primary 32D05, 32H35, 32H40, 32M05, 32T05
• DOI: https://doi.org/10.1090/tran/6690
• MathSciNet review: 3557783