Comptes Rendus
Differential geometry
A gap theorem for minimal submanifolds in Euclidean space
Comptes Rendus. Mathématique, Volume 353 (2015) no. 2, pp. 173-177.

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

On démontre que, pour une sous-variété minimale complète Mn immergée dans l'espace euclidien Rn+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)ε pour tous xM, où ε est une constante positive ne dépendant que de n, alors M est un sous-espace affine de Rn+d.

Published online:
DOI: 10.1016/j.crma.2014.11.009

Entao Zhao 1, 2; Shunjuan Cao 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
     author = {Entao Zhao and Shunjuan Cao},
     title = {A gap theorem for minimal submanifolds in {Euclidean} space},
     journal = {Comptes Rendus. Math\'ematique},
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Entao Zhao; Shunjuan Cao. A gap theorem for minimal submanifolds in Euclidean space. Comptes Rendus. Mathématique, Volume 353 (2015) no. 2, pp. 173-177. doi : 10.1016/j.crma.2014.11.009.

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