Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

Request Permissions   Purchase Content 
 

 

Proper holomorphic mappings between invariant domains in $ \mathbb{C}^n$


Authors: Jiafu Ning, Huiping Zhang and Xiangyu Zhou
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
Published electronically: May 6, 2016
MathSciNet review: 3557783
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 32D05, 32H35, 32H40, 32M05, 32T05

Retrieve articles in all journals with MSC (2010): 32D05, 32H35, 32H40, 32M05, 32T05


Additional Information

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
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
Email: xyzhou@math.ac.cn

DOI: https://doi.org/10.1090/tran/6690
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
Article copyright: © Copyright 2016 American Mathematical Society