Embedding and surrounding with positive mean curvature
type: Item
instance of: scholarly article
title: Embedding and surrounding with positive mean curvature
zbMATH Open document ID: 0555.53027
DOI: 10.1007/BF01388830
author: Marie-Louise Michelsohn
author: H. Blaine jun. Lawson
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143152
review text: This article divides into two parts: a) Let \(M\) be a compact connected manifold with non-empty boundary \(\partial M\). According to a theorem of Gromov, \(M\) carries a Riemannian metric of strictly positive sectional curvature \(K\). The authors investigate under which topological assumptions one can achieve, in addition, that \(\partial M\) has positive mean curvature \(H\). Theorem: \(M (\dim M\neq 4)\) carries a Riemannian metric with \(K>0\) and \(H>0\) iff \(M\) is 1-thin. b) Let \(N\) be a closed manifold without boundary and \(f: N\to \bar M\) a (normalized) hypersurface embedding into an arbitrary Riemannian manifold \(\bar M\). Under which conditions is the embedding of positive mean curvature? Theorem: If \(dim \bar M\neq 4\) and \(f\) bounds a compact domain \(D_ f\) in \(\bar M,\) then \(f\) is strongly isotopic to an embedding of positive mean curvature if \(D_ f\) is 1-thin. The basic tool is a handle-attaching theorem stating a method for constructing new hypersurfaces with \(H>0\) out of a method for constructing new hypersurfaces with \(H>0\) out of given ones by attaching ambient handles of codimension \(\geq 2\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 3884911
zbMATH Keywords: bridge principle
zbMATH Keywords: positive sectional curvature
zbMATH Keywords: mean curvature
zbMATH Keywords: handle- attaching theorem
reviewed by: Wolf Struebing
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3884911