Boundary regularity of correspondences in \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Boundary regularity of correspondences in \(\mathbb{C}^2\)
zbMATH Open document ID: 0939.32014
DOI: 10.1007/PL00004728
author: Kaushal Verma
published in: Mathematische Zeitschrift
publication date: 2000-07-02T00:00:00Z
review text: It has been previously proved by \textit{K. Diederich} and \textit{S. Pinchuk} [Indiana Univ. Math. J. 44, No. 4, 1089-1126 (1995; Zbl 0857.32015)], that a proper holomorphic mapping between two bounded domains with smooth real analytic boundaries in \(\mathbb C^2\) extends holomorphically to an open neighbourhood crossing the boundary. The present article attempts to answer a similar question for proper holomorphic correspondence between domains in \(\mathbb C^2\). The main theorem asserts the following: Let \(\widehat f: D\to D'\) be a proper holomorphic correspondence between domains \( D\) and \(D'\) in \(\mathbb C^2\) with inverse correspondence \(\widehat g\), and let \(\Gamma_1\subset\subset \Gamma \subset \partial D\) and \(\Gamma' \subset \partial D'\) be open pieces of the boundaries such that \(\partial D\) (resp. \(\partial D'\)) is real analytic and of finite type in an open neighbourhood of \(\bar \Gamma\) (resp. \(\bar \Gamma'\)). Assume that the cluster point of \(\Gamma_1\) under \(\widehat f\) is a subset of \(\Gamma'\) and the cluster point of \(\Gamma'\) under \(\widehat g\) is a subset of \(\Gamma\). Then \(\widehat f\) extends as a holomorphic correspondence near each \(z_0\in \Gamma_1\).
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 32H40
zbMATH DE Number: 1336887
zbMATH Keywords: proper holomorphic mappings
zbMATH Keywords: holomorphic correspondence
zbMATH Keywords: boundary regularity
zbMATH Keywords: pseudoconvex domains
reviewed by: Genkai Zhang
MaRDI profile type: Publication
full work available at URL: pl00004728
OpenAlex ID: W3048971710
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description: scientific article; zbMATH DE number 1336887