The generalized Lu rigidity theorem for submanifolds with parallel mean curvature
type: Item
instance of: scholarly article
title: The generalized Lu rigidity theorem for submanifolds with parallel mean curvature
zbMATH Open document ID: 1396.53069
author: Yan Leng
author: Hongwei Xu
published in: Manuscripta Mathematica
publication date: 2018-01-26T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional oriented compact submanifold with parallel mean curvature in the unit sphere \(S^{n+p}\). Let \(\lambda_2\) the second largest eigenvalue of the fundamental matrix of \(M\) and let \[ \alpha(n,H)=n +\frac{n^3 H^2}{2(n-1)} -\frac{n(n-2)H}{2(n-1)}\sqrt{n^2 H^2 +4(n-1)} \] where \(H\) is the mean curvature of \(M\). Let \(S\) be the squared length of the second fundamental form \(h\) of \(M\). If \(S +\lambda_2 \leq \alpha(n,H)\), then the authors obtain a rigidity theorem for \(M\), which generalizes Lu's optimal rigidity theorem for minimal submanifolds in the unit sphere \(S^{n+p}\).
reviewed by: Abdoul Salam Diallo
Mathematics Subject Classification ID: 53C24
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 6830034
zbMATH Keywords: submanifolds
zbMATH Keywords: mean curvature
MaRDI profile type: Publication
full work available at URL: s00229-017-0932-9
OpenAlex ID: W2774929669
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DOI: 10.1007/S00229-017-0932-9
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description: scientific article; zbMATH DE number 6830034