A generalization of Cartan's theorem to proper holomorphic mappings
type: Item
instance of: scholarly article
title: A generalization of Cartan's theorem to proper holomorphic mappings
zbMATH Open document ID: 0638.32025
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1988-01-01T00:00:00Z
review text: The theorem of \textit{H. Cartan} [Math. Z. 35, 760-773 (1932; Zbl 0004.40602)] says: Consider a sequence of biholomorphic mappings \(f_ i:\Omega_ 1\to \Omega_ 2\) between bounded domains in \({\mathbb{C}}^ n,\) which converges uniformly on compact subsets of \(\Omega_ 1\) to a mapping f. Then: (A) f is a biholomorphic mapping of \(\Omega_ 1\) onto \(\Omega_ 2\), iff (B) \(f(\Omega_ 1)\) is not a subset of \(b\Omega_ 2\), iff (C) the jacobian determinant det [f'(z)] is not identically zero on \(\Omega_ 1.\) The author gives an example in \({\mathbb{C}}^ 1 \)to show that one can not replace ``biholomorphic'' by ``proper'' in this theorem, and proves a theorem in which under some additional hypotheses this is possible. The author's theorem says: Let \(\Omega_ 1\) and \(\Omega_ 2\) be additionally pseudoconvex and have bounded plurisubharmonic exhaustion functions, and let \(f_ i\) be proper holomorphic and additionally have multiplicities bounded by some m. Then : (A') f is a proper mapping of \(\Omega_ 1\) onto \(\Omega_ 2\), iff Cartan's (B) holds, iff Cartan's (C) holds. Moreover, if (A') takes place, then the multiplicity of f is not larger than m. All the necessary notions are defined in the text. The author describes how to use the theorem to examine which domains in \({\mathbb{C}}^ n \)have no proper holomorphic self-mappings which are not biholomorphic. He also asks (a) if his theorem holds for bounded taut domains, and (b) if the exhaustion functions are really needed in his theorem.
Mathematics Subject Classification ID: 32H30
Mathematics Subject Classification ID: 32H25
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 4040302
zbMATH Keywords: sequence of proper holomorphic mappings converging uniformly on compact subsets
zbMATH Keywords: multiplicity of a mapping
zbMATH Keywords: pseudoconvex
zbMATH Keywords: plurisubharmonic exhaustion functions
author: Steven R. Bell
reviewed by: Krzysztof Dabrowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4040302