Some remarks about Reinhardt domains in \(\mathbb{C}^n\)
type: Item
instance of: scholarly article
title: Some remarks about Reinhardt domains in \(\mathbb{C}^n\)
zbMATH Open document ID: 1033.32007
author: Nguyen Quang Dieu
author: Hai Mau Le
published in: Illinois Journal of Mathematics
publication date: 2003-10-16T00:00:00Z
review text: The main result of this note is an analogue of the well known fact that every holomorphic function on a neighbourhood of the closure of the Hartogs triangle extends holomorphically to a neighbourhood of the closure of the unit bidisk. It is shown that, given a bounded Reinhardt domain \(D\subset\mathbb C^n\), there exists a hyperconvex domain \(\Omega\) such that \(\Omega\) contains \(D\) and every holomorphic function on a neighbourhood of \(\overline D\) extends to a neighbourhood of \(\overline\Omega\). The connection between the Carathéodory hyperbolicity of a Reinhardt domain and that of its envelope of holomorphy is studied. An example of a Carathéodory hyperbolic Reinhardt domain in \(\mathbb C^3\) such that its envelope of holomorphy contains a complex line and hence is not Carathéodory hyperbolic, is given. However, it is shown that such an example cannot exist in \(\mathbb C^2\). An explicit description of a Stein neighbourhoods basis for the closure of a hyperconvex Reinhardt domain is obtained.
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32E20
zbMATH DE Number: 1993663
zbMATH Keywords: Reinhardt domain
zbMATH Keywords: Hartogs triangle
zbMATH Keywords: Stein neighbourhood basis
zbMATH Keywords: envelope of holomorphy
zbMATH Keywords: Carathéodory hyperbolicity of Reinhardt domains
zbMATH Keywords: holomorphic extension
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1993663