On the existence of bounded holomorphic functions on complete Kähler manifolds
type: Item
instance of: scholarly article
title: On the existence of bounded holomorphic functions on complete Kähler manifolds
zbMATH Open document ID: 0588.32001
DOI: 10.1007/BF01388588
author: John S. Bland
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143272
review text: Sufficient conditions are given for the existence of bounded (non- constant) holomorphic functions on a complete Kähler manifold \(M^ n\), \(n\geq 2\). Modified exponential coordinates and asymptotic conditions on the curvature are used to determine the asymptotic behavior of the metric and the J operator. This leads to a smooth CR structure on \(\partial M\) (diffeomorphic to \(\partial {\mathbb{B}}^{2n})\) for which CR functions on \(\partial M\) extend to holomorphic functions on M. For \(n\geq 3\), an imbedding theorem of \textit{L. Boutet de Monvel} [Sémin. Goulaouic-Lion- Schwartz 1974-1975, Exposé IX, 13 p. (1975; Zbl 0317.58003)] provides smooth non-constant CR functions on \(\partial M\). Thus, bounded holomorphic functions exist for \(M^ n\), \(n\geq 3\), a complete Kähler manifold with non-positive sectional curvature under some condition for suitable exponential coordinates.
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32E35
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 3944252
zbMATH Keywords: bounded holomorphic functions on complete Kähler manifolds
zbMATH Keywords: extension of CR functions to holomorphic functions
zbMATH Keywords: CR structure
MaRDI profile type: Publication
cites work: Function theory on manifolds which possess a pole
cites work: Q4163752
cites work: Harmonic maps of manifolds with boundary
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cites work: Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems
description: scientific article; zbMATH DE number 3944252