On completeness of invariant metrics of Reinhardt domains
type: Item
instance of: scholarly article
title: On completeness of invariant metrics of Reinhardt domains
zbMATH Open document ID: 0815.32001
DOI: 10.1007/BF01189891
author: Siqi Fu
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: Let \(\Omega\) be a bounded Reinhardt domain of holomorphy with center 0 in \(\mathbb{C}^ n\). The author proves that \(\Omega\) is complete (Kobayashi) hyperbolic and that if \(\Omega\) meets a coordinate hyperplane \(\{z_ j = 0\}\) whenever its closure does \((j = 1, \dots, n)\), then all closed balls for the Carathéodory distance are compact. This generalizes a result of \textit{P. Pflug} [Functional analysis, holomorphy and approximation theory. II, Proc. Semin., Rio de Janeiro 1981, North Holland Math. Stud. 86, 331- 337 (1984; Zbl 0536.32001)], who assumed that \(0 \in \Omega\). If \(\Omega\) has \({\mathcal C}^ 1\) boundary, the intersection condition is automatically satisfied, and it follows from a theorem of \textit{T. W. Gamelin} [Math. Ann. 238, No. 2, 131-139 (1978; Zbl 0372.46050)] that the boundary of \(\Omega\) is the union of analytic discs and the set of peak points for the algebra of uniform limits of holomorphic functions on the closure of \(\Omega\).
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 687059
zbMATH Keywords: bounded Reinhardt domain
zbMATH Keywords: Carathéodory distance
zbMATH Keywords: Kobayashi distance
zbMATH Keywords: uniform algebra
zbMATH Keywords: domain of holomorphy
zbMATH Keywords: peak points
reviewed by: Theodore J. Barth
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 687059