A gap theorem for minimal submanifolds in Euclidean space

Zhao, Entao; Cao, Shunjuan

doi:10.1016/j.crma.2014.11.009

Differential geometry

A gap theorem for minimal submanifolds in Euclidean space
[Un théorème de seuil pour les sous-variétés minimales dans l'espace euclidien]

Présenté par : the Editorial Board

Zhao, Entao ^1,² ; Cao, Shunjuan ³

¹ Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People's Republic of China
² National Center for Theoretical Sciences, Taipei Office, Taipei, 10617, Taiwan
³ Department of Mathematics, Zhejiang Agricultural and Forestry University, Lin'an, 311300, People's Republic of China

Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 173-177.

Résumé
Abstract

On démontre que, pour une sous-variété minimale complète $M^{n}$ immergée dans l'espace euclidien $R^{n + d}$ , si la seconde forme fondamentale A et la fonction distance intrinsèque r mesurée à partir d'un point fixe satisfont l'inégalité $r (x) | A | (x) \leq ε$ pour tous $x \in M$ , où ε est une constante positive ne dépendant que de n, alors M est un sous-espace affine de $R^{n + d}$ .

We prove that for a complete minimal submanifold $M^{n}$ immersed in the Euclidean space $R^{n + d}$ , if the second fundamental form A and the intrinsic distance function r from a fixed point satisfy $r (x) | A | (x) \leq ε$ for all $x \in M$ , where ε is a positive constant depending only on n, then M is an affine subspace of $R^{n + d}$ .

Reçu le : 2014-09-03
Accepté le : 2014-11-12
Publié le : 2014-12-30

DOI : 10.1016/j.crma.2014.11.009

Affiliations des auteurs :

Zhao, Entao ^{1, 2} ; Cao, Shunjuan ³

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     title = {A gap theorem for minimal submanifolds in {Euclidean} space},
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Zhao, Entao; Cao, Shunjuan. A gap theorem for minimal submanifolds in Euclidean space. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 173-177. doi : 10.1016/j.crma.2014.11.009. http://www.numdam.org/articles/10.1016/j.crma.2014.11.009/

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