Holomorphic equivalence problem for a certain class of unbounded Reinhardt domains in \(\mathbb{C}^ 2\)
type: Item
instance of: scholarly article
title: Holomorphic equivalence problem for a certain class of unbounded Reinhardt domains in \(\mathbb{C}^ 2\)
zbMATH Open document ID: 0786.32001
author: Satoru Shimizu
published in: Osaka Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: An answer to the holomorphic equivalence problem for arbitrary bounded Reinhardt domains was given by the author in Tohoku Math. J., II. Ser. 40, No. 1, 119-152 (1988; Zbl 0646.32003). It seems interesting to investigate whether a similar result on unbounded Reinhardt domains holds or not. Now, for a pair \((a,b)\) of nonnegative real constants with \((a,b) \neq(0,0)\) and a positive real constant \(r\), let us consider an unbounded Reinhardt domain \(D_{a,b}(r)\) in \(\mathbb{C}^ 2\) given by \[ D_{a,b}(r)=\{ (z,w) \in \mathbb{C}^ 2 \mid | z |^ a | w |^ b<r \}. \] Here, when \(ab=0\), for example, when \(b=0\), the domain \(D_{a,0}(r)\) is understood as \[ D_{a,0}(r)=\{(z,w) \in \mathbb{C}^ 2 \mid | z |^ a<r \}. \] We are concerned with the holomorphic automorphisms and the equivalence of the domains \(D_{a,b}(r)\). In the present paper, we confine ourselves to the case where \(a\) and \(b\) are integers. The case where \(a\) and \(b\) are arbitrary nonnegative real constants will be treated in the subsequent paper [part II of this paper, same title (preprint)]. Our main result of this paper can be stated as the following theorem. Theorem. If a domain \(D_{a,b}(r)\) with \((a,b) \in(\mathbb{Z}_{\geq 0})^ 2\) is biholomorphic to a domain \(D_{u,v}(s)\) with \((u,v) \in(\mathbb{Z}_{\geq 0})^ 2\), then there exists a transformation \(\varphi\) given by \[ \varphi:\mathbb{C}^ 2 \ni(z,w) \mapsto (\alpha z,\beta w) \in \mathbb{C}^ 2 \] or \[ \varphi:\mathbb{C}^ 2 \ni(z,w) \mapsto (\gamma w,\delta z) \in \mathbb{C}^ 2 \] such that \(\varphi (D_{a,b} (r))=D_{u,v}(s)\), where \(\mathbb{Z}_{\geq 0}\) denotes the set of nonnegative integers and \(\alpha,\beta,\gamma,\delta\) are nonzero complex constants.
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 59709
zbMATH Keywords: Liouville foliation
zbMATH Keywords: unbounded Reinhardt domains
zbMATH Keywords: automorphisms
zbMATH Keywords: equivalence
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 59709