Proper holomorphic mappings in some class of unbounded domains
type: Item
instance of: scholarly article
title: Proper holomorphic mappings in some class of unbounded domains
zbMATH Open document ID: 0984.32002
author: Armen Edigarian
author: Włodzimierz Zwonek
published in: Kodai Mathematical Journal
publication date: 2002-05-02T00:00:00Z
review text: Elementary Reinhardt domains in \(\mathbb C^n\) are defined as the domains of the form \(D_\alpha = \{ z=(z_1,\dots,z_n)\in \mathbb C^n : |z_1|^{\alpha_1} \cdots |z_n|^{\alpha_n} < 1 \}\), where \(\alpha := (\alpha_1, \dots, \alpha_n) \in \mathbb R_+^n\) is said to be of rational type if there exists \(\lambda >0\) such that \(\lambda \alpha = (\alpha'_1, \dots, \alpha'_n)\) where the \(\alpha'_j\) are relatively prime natural numbers. Let \(F\) be a proper holomorphic mapping from \(D_\alpha\) to \(D_\beta\), two elementary Reinhardt domains. Then \(\alpha\) and \(\beta\) must have the same type (rational or irrational). If \(n=2\), the authors give an explicit form for \(F\) depending on a discrete set of parameters and a zero-free holomorphic function on the unit disk (the coordinates of \(F\) must be certain quotients of monomials multiplied or divided by this function). If \(n\) is arbitrary and \(F\) is furthermore biholomorphic, they prove that both domains have to be linearly biholomorphic. Those results improve on previous work of \textit{S. Shimizu} [Osaka J. Math. 28, No. 3, 609-621 (1991; Zbl 0786.32001); Kodai Math. J. 15, No. 3, 430-444 (1992; Zbl 0790.32001)]. The method of proof involves the pluricomplex Green function for several poles [\textit{P. Lelong}, J. Math. Pures Appl., IX. Sér. 68, No. 3, 319-347 (1989; Zbl 0633.32019)], which is invariant under proper holomorphic mappings. The authors compute this function for elementary Reinhardt domains using previous results for the Green function with one pole [\textit{M. Jarnicki} and \textit{P. Pflug}, `Invariant distances and metrics in complex analysis'. De Gruyter Expositions in Mathematics. 9. Berlin: Walter de Gruyter (1993; Zbl 0789.32001); and \textit{P. Pflug} and \textit{W. Zwonek}, Ann. Pol. Math. 69, No. 2, 175-196 (1998); erratum ibid. No. 3, 301 (1998; Zbl 0926.32015)]. This function is constant on the level sets of the function \(z^\alpha\) (which sends \(D_\alpha\) to the unit disk), s…
reviewed by: Pascal J. Thomas
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A05
Mathematics Subject Classification ID: 32U35
zbMATH DE Number: 1437241
zbMATH Keywords: proper holomorphic mappings
zbMATH Keywords: Reinhardt domains
zbMATH Keywords: pluricomplex Green functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044089
OpenAlex ID: W2015032579
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DOI: 10.2996/KMJ/1138044089
description: scientific article; zbMATH DE number 1437241