A pinching theorem for minimal submanifolds in a sphere
type: Item
instance of: scholarly article
title: A pinching theorem for minimal submanifolds in a sphere
zbMATH Open document ID: 0762.53041
DOI: 10.1007/BF01196863
published in: Archiv der Mathematik
publication date: 1992-09-27T00:00:00Z
review text: This paper deals with the pinching problem for the length square of the second fundamental form of a compact minimal submanifold in a sphere. The main results are as follows: Let \(M\) be a compact \(n\)-dimensional minimal submanifold in a unit sphere \(S^{n+p}\) of dimension \(n+p\). Denote by \(S\) the length square of the second fundamental form of \(M\). If \(n\) is even and \(S\leq n/[1+2(n-1)/(3n-2)]\), then \(M\) is either totally geodesic or a Veronese surface in \(S^ 4\). Moreover, the case where \(n\) is odd is considered similarly. Recently, a better uniform pinching constant was shown by \textit{Li Anmin} and \textit{Li Jimin} [see Arch. Math. 58, 582-594 (1992; Zbl 0731.53056)].
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 62241
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: second fundamental form
zbMATH Keywords: Veronese surface
author: Changyu Xia
MaRDI profile type: Publication
cites work: Q5341449
cites work: Q5678153
cites work: Minimal Submanifolds of a Sphere With Bounded Second Fundamental Form
cites work: Three-manifolds with positive Ricci curvature
cites work: Minimal varieties in Riemannian manifolds
cites work: Minimal immersions of 2-manifolds into spheres
full work available at URL: https://doi.org/10.1007/bf01196863
OpenAlex ID: W2330187814
description: scientific article; zbMATH DE number 62241