The classification of proper holomorphic mappings between special Hartogs triangles of different dimensions
type: Item
instance of: scholarly article
title: The classification of proper holomorphic mappings between special Hartogs triangles of different dimensions
zbMATH Open document ID: 1168.32017
author: Zhihua Chen
author: Yang Liu
published in: Chinese Annals of Mathematics. Series B
publication date: 2008-10-27T00:00:00Z
review text: The proper holomorphic mapping theory dates from the 1950s, and there are many good results on it. The classification of proper holomorphic mappings is an important and difficult problem, especially between bounded domains of different dimensions. By a classical result of \textit{H. Alexander} [Math. Ann. 209, 249--256 (1974; Zbl 0272.32006)], every proper holomorphic self-mapping of \(B^{n}\) with \(n\geq 2\) is equivalent to the identity mapping. In 2001, \textit{X. Huang} and \textit{S. Ji} [Invent. Math. 145, No. 2, 219--250 (2001; Zbl 1038.32018)] proved that there are only two equivalence classed in \(\text{Rat}(B^{n},B^{N})\) for \(n>2\). In 2006, \textit{X. Huang, D. Xu} and \textit{S. Ji} [Math. Res. Lett. 13, No. 4, 515--529 (2006; Zbl 1158.32007)] found a new gap phenomenon for proper holomorphic mappings from \(B^{n}\) to \(B^{N}\) when \(N\leq 3n-4\). In this paper, the authors consider the special Hartogs triangles defined as follows: \[ \begin{aligned} \Omega(n_{1},m_{1})&= \bigg\{(z,w)\in\mathbb C^{n_{1}+m_{1}}: 0<\sum_{i=1}^{n_{1}}|z_{i}|^{2}<\sum_{j=1}^{m_{1}}|w_{j}|^{2}<1 \bigg\}\\ \Omega(n_{2},m_{2})&= \bigg\{(z',w')\in\mathbb C^{n_{2}+m_{2}}: 0<\sum_{i=1}^{n_{2}}|z_{i}'|^{2}<\sum_{j=1}^{m_{2}}|w_{j}'|^{2}<1\bigg\}. \end{aligned} \] They discuss proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem. The main idea is from [\textit{Z. H. Chen}, and \textit{Y. C. Liu}, Chin. J. Contemp. Math. 24, No.~3, 215--220 (2003; Zbl 1069.32500)].
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 32A22
zbMATH DE Number: 5357883
zbMATH Keywords: proper holomorphic mapping
zbMATH Keywords: Hartogs triangle
zbMATH Keywords: automorphism group
MaRDI profile type: Publication
full work available at URL: s11401-007-0044-z
OpenAlex ID: W2032205076
cites work: Mapping \(B^n\) into \(B^{2n-1}\)
cites work: A new gap phenomenon for proper holomorphic mappings from \(\mathbb B^n\) into \(\mathbb B^N\)
cites work: On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions.
cites work: Q4172303
cites work: Q4679724
cites work: Rigidity of proper self-mapping on some kinds of generalized Hartogs triangle
cites work: Proper holomorphic mappings between some nonsmooth domains
cites work: Proper holomorphic maps from domains in \(\mathbb C^{2}\) with transverse circle action
DOI: 10.1007/S11401-007-0044-Z
description: scientific article; zbMATH DE number 5357883