On the structure of complete Kähler manifolds with nonnegative curvature near infinity
type: Item
instance of: scholarly article
title: On the structure of complete Kähler manifolds with nonnegative curvature near infinity
zbMATH Open document ID: 0695.53052
DOI: 10.1007/BF01234432
author: Peter Li
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143771
review text: The main purpose of this article is to study the geometric structure at infinity and the topology of a complete noncompact Kähler manifold M whose sectional curvature is nonnegative outside a compact subset (or for the sake of convenience near infinity). More precisely, the fact is established that if M is of complex dimension m and has more than one end, then except on a compact set, M is metrically a product of a complete complex 1-manifold \(\Sigma\) with a compact Kähler manifold N. The manifold \(\Sigma\) has nonnegative curvature near infinity and hence must be conformally equivalent to a Riemann surface with finite punctures. The compact manifold N has nonnegative sectional curvature and hence its universal cover \(\tilde N\) must be isometrically a product of Kähler manifolds which are either i) isometrically \({\mathbb{C}}^ k\), ii) isometrically an irreducible compact Hermitian symmetric space of rank not less than 2, or iii) isometrically a \({\mathbb{C}}{\mathbb{P}}^ k\) with a nonnegatively curved Kähler metric. The main technique is to utilize the nonconstant positive harmonic functions. The cases when M has more than one end with one of it being a large end and if M has one large end are considered. The first homology group of M is studied and the first Betti number is estimated from above by 2m-1, where m is the complex dimension of M. In addition, if M has positive sectional curvature near infinity and has a large end, then its first homology group must be trivial.
reviewed by: Neda Bokan
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 4138717
zbMATH Keywords: finite topological type
zbMATH Keywords: harmonic function
zbMATH Keywords: Buseman function
zbMATH Keywords: geometric structure at infinity
zbMATH Keywords: Kähler manifold
zbMATH Keywords: nonnegative curvature
zbMATH Keywords: first homology group
zbMATH Keywords: first Betti number
zbMATH Keywords: large end
MaRDI profile type: Publication
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OpenAlex ID: W2006789637
description: scientific article; zbMATH DE number 4138717