Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin
type: Item
instance of: scholarly article
title: Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin
zbMATH Open document ID: 0601.32030
DOI: 10.1007/BF02566366
author: D. Barret
published in: Commentarii Mathematici Helvetici
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/139990
review text: Bounded Reinhardt domains \(D\subset {\mathbb{C}}^ n\) satisfying the following condition (*) are considered in the paper: (*) There is an integer \(k=k(D)\), \(0\leq k\leq n\), such that \(D\cap \{z_ j=0\}=\emptyset\) for \(j=1,...,k\) and \(\bar D\cap \{z_ j=0\}=\emptyset\) for \(j=k+1,...,n.\) Theorem 1. If \(D_ 1\) and \(D_ 2\) are holomorphically equivalent bounded Reinhardt domains in \({\mathbb{C}}^ n\) satisfying (*), then \(k(D_ 1)=k(D_ 2)\), and there is a biholomorphic map \(F: D_ 1\to D_ 2\) of the form \(F(z',z'')=(c_ 1z^{''\beta_ 1}z_{\sigma (1)},...,c_ kz^{''\beta_ k}z_{\sigma (k)},\quad c_{k+1}z^{''\alpha_{k+1}},...,c_ nz^{''\alpha_ n})\) where \(k=k(D_ 1)=k(D_ 2)\), \(z''=(z_{k+1},...,z_ n)\), \(\beta_ j\in {\mathbb{Z}}^{n-k}\), \((\alpha_{k+1},...,\alpha_ n)^ t\in GL(n- k,{\mathbb{Z}})\), \(\sigma\) is a permutation of \(\{\) 1,...,k\(\}\) and \(c_ 1,...,c_ n\) are positive constants. For domains containing the origin this was proved by \textit{T. Sunada} [Math. Ann. 235, 111-128 (1978; Zbl 0357.32001)]. In the case when \(k(D_ 1)=k(D_ 2)=0\) Theorem 1 may be viewed as a special case of a \textit{E. Bedford}'s result [Pac. J. Math. 87, 271-281 (1980; Zbl 0449.32024)]. Theorem 2. If \(F: D_ 1\to D_ 2\) is a proper holomorphic map of bounded Reinhardt domains in \({\mathbb{C}}^ n\) and if \(D_ 1\) satisfies (*), then F extends holomorphically to a neighborhood of \(\bar D{}_ 1.\) For complete domains this result was proved by \textit{S. Bell} [Trans. Am. Math. Soc. 270, 685-691 (1982; Zbl 0482.32007)].
Mathematics Subject Classification ID: 32H99
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32D15
zbMATH DE Number: 3969068
zbMATH Keywords: holomorphic extension
zbMATH Keywords: holomorphically equivalent bounded Reinhardt domains
zbMATH Keywords: proper holomorphic map
reviewed by: Johann Davidov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3969068