Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps
type: Item
instance of: scholarly article
title: Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps
zbMATH Open document ID: 0818.32008
DOI: 10.1007/BF02921586
author: J. E. Joseph
author: Myung H. Kwack
published in: The Journal of Geometric Analysis
publication date: 1995-08-15T00:00:00Z
review text: Let \(X\) be a (not necessarily relatively compact) complex subspace of the complex space \(Y\). Let \(F_ X\) denote the Kobayashi-Royden pseudometric on \(X\), and let \(E\) be a length function on \(Y\) [in the sense of \textit{S. Lang}, `Introduction to complex hyperbolic spaces' (1987; Zbl 0628.32001)]. Following an idea of \textit{H. L. Royden} [Several complex variables. II, Conf. Univ. Maryland 1970, 125-137 (1971; Zbl 0218.32012)], the authors say that a point \(p \in \overline X\) is a hyperbolic point if there are a neighborhood \(U\) of \(p\) in \(Y\) and a constant \(c > 0\) such that \(F_ X \geq cE\) on \(U \cap X\); they observe that \(X\) is hyperbolically imbedded in \(Y\) if and only if every point of \(\overline X\) is a hyperbolic point. The authors then use various spaces of (limits of) holomorphic maps to characterize hyperbolic points and, thereby, hyperbolic embeddedness. These lead to generalizations of results previously known when \(Y\) is nonsingular or \(X = Y\) or \(X \Subset Y\). Included are theorems of \textit{M. Abate} [Proc. Am. Math. Soc. 117, No. 3, 789-793 (1993; Zbl 0773.32018)], \textit{P. Kiernan} [Math. Ann. 204, No. 3, 203-209 (1973; Zbl 0244.32010)], the second author [Ann. Math., II. Ser. 90, No. 1, 9-22 (1969; Zbl 0179.12103)], and, most notably, \textit{J. Noguchi} [Publ. Res. Inst. Math. Sci. 21, No. 1, 27-46 (1985; Zbl 0583.32061)], where the authors' new proof avoids measure theory.
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 32H25
zbMATH DE Number: 653343
zbMATH Keywords: divisor with normal crossings
zbMATH Keywords: hyperbolically imbedded complex space
zbMATH Keywords: hyperbolic point
zbMATH Keywords: Kobayashi pseudodistance
zbMATH Keywords: Kobayashi-Royden pseudometric
reviewed by: Theodore J. Barth
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921586
OpenAlex ID: W2062179395
description: scientific article; zbMATH DE number 653343