Rigidity of stable minimal hypersurfaces
type: Item
instance of: scholarly article
title: Rigidity of stable minimal hypersurfaces
zbMATH Open document ID: 0957.53031
author: Ying Shen
author: Shun-Hui Zhu
published in: Mathematische Annalen
publication date: 1997-09-09T00:00:00Z
review text: In this interesting paper, the authors -- motivated by \textit{D. Fisher-Colbrie} and \textit{R. Schoen} [Common. Pure Appl. Math. 33, 199-211 (1980; Zbl 0439.53060)] and \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math. Stud. 102, 209-228 (1982; Zbl 0481.53036)] -- study stability of minimal hypersurfaces assuming, on the ambient manifold, that its dimension is at least 4 and a natural condition on its sectional curvature, and the completeness of the hypersurface. Concretely, they prove that if \(\Sigma^n=N\times {\mathbb R}\) is a complete stable minimal hypersurface in \(M^{n+1}\), where \(N\) is a compact manifold admitting a metric of nonpositive sectional curvature, \(n=3\) or \(n=4\) and the sectional curvature of \(M\) is nonnegative, then \(\Sigma ^n\) is flat and totally geodesic. When \(n=2\) this result was proved in the paper cited above, under the nonnegative scalar curvature condition. The authors conjecture that this result can be true for all dimensions. Under similar conditions, they also prove a global splitting theorem for this case.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1058821
zbMATH Keywords: minimal hypersurface
zbMATH Keywords: stability
zbMATH Keywords: sectional curvature
zbMATH Keywords: totally geodesic
zbMATH Keywords: scalar curvature
MaRDI profile type: Publication
full work available at URL: s002080050105
OpenAlex ID: W2054511687
DOI: 10.1007/S002080050105
description: scientific article; zbMATH DE number 1058821