Rigidity of compact minimal submanifolds in a unit sphere
type: Item
instance of: scholarly article
title: Rigidity of compact minimal submanifolds in a unit sphere
zbMATH Open document ID: 0770.53040
DOI: 10.1007/BF01667404
author: S. Singh
published in: Geometriae Dedicata
publication date: 1993-04-01T00:00:00Z
review text: The authors prove the following result: Let \(M\) be a smooth \(n\)- dimensional compact minimal submanifold of a unit sphere of codimension \(p\geq 2\), and denote by \(S\) the square of the length of the second fundamental form of this submanifold. If \(S\leq{2\over 3}n\) on \(M\), then \(M\) is totally geodesic or a Veronese surface. This result was already proved by \textit{Li An-Min} and \textit{Li Jimin} [Arch. Math. 58, 582-594 (1992)]. All authors use the formula for \(S\) of \textit{J. Simons} [Ann. Math., II. Ser. 88, 62-105 (1968; Zbl 0181.497)], but give different proofs for some subtil estimates.
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 149052
zbMATH Keywords: totally geodesic
zbMATH Keywords: Veronese surface
reviewed by: Udo Simon
MaRDI profile type: Publication
cites work: Minimal varieties in Riemannian manifolds
cites work: Submanifolds with Constant Mean Curvature II
cites work: Addendum to my paper ''On veronese manifolds''
cites work: On intrinsic rigidity for minimal submanifolds in a sphere
cites work: Minimal Submanifolds of a Sphere With Bounded Second Fundamental Form
cites work: Eigenvalues of the Laplacian and curvature
full work available at URL: https://doi.org/10.1007/bf01667404
OpenAlex ID: W2081034517
description: scientific article; zbMATH DE number 149052