Removable singularities and Liouville-type property of analytic multivalued functions
type: Item
instance of: scholarly article
title: Removable singularities and Liouville-type property of analytic multivalued functions
zbMATH Open document ID: 0777.32006
author: Tran Ngoc Giao
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1993-12-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73303
review text: The main results of the paper are the following two theorems. 1) Let \(S\) be a closed subset of an open set \(G\subset\mathbb{C}^ n\) and let \(Y\) be a Stein space. Suppose that \(K:G\backslash S\to F_ c(Y)\) is an analytic multivalued function \((F_ c(Y):=\{A\subset Y:A\) is a compact, \(A\neq\emptyset\})\). Then \(K\) extends analytically to \(G\) if one of the following conditions is satisfied. a) \(S=H\cap(G\backslash U)\), where \(H\) is an analytic set in \(G\) and \(U\) is an open subset of \(G\) such that \(U\) intersects any connected component of \(H\). b) \(H\) is of \((2n-2)\)-Hausdorff measure zero. c) \(S\) is pluripolar and \(K\) is locally compact, i.e. each point \(x\in G\) has an open neighbourhood \(U\) such that \(K(U\backslash S)\) is relatively compact. 2) Let \(D\) be a convex domain in \(\mathbb{C}^ n\). The following conditions are equivalent: (i) For every analytic multivalued function \(K:\mathbb{C}\to F_ e(D)\) the function \(\hat K(x):=\) the polynomially convex hull of \(K(x)\), \(x\in\mathbb{C}\), is constant. (ii) Every analytic multivalued function \(K:\Delta\backslash\{0\}\to F_ c(D)\) extends analytically to \(\Delta\), where \(\Delta\) denotes the unit disc. (iii) For arbitrary polar set \(S\subset\Delta\) every analytic multivalued function \(K:\Delta\backslash S\to F_ c(D)\) extends analytically to \(\Delta\).
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32D20
Mathematics Subject Classification ID: 30G30
zbMATH DE Number: 202539
zbMATH Keywords: Liouville-type property
zbMATH Keywords: analytic multivalued functions
zbMATH Keywords: removable singularities
MaRDI profile type: Publication
OpenAlex ID: W2050401359
cites work: Convex Domains and Kobayashi Hyperbolicity
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DOI: 10.5802/AFST.748
description: scientific article; zbMATH DE number 202539