Global pinching theorems for minimal submanifolds in spheres

Kairen Cai¹

¹ Department of Mathematics Hangzhou Teachers' College 96 Wen Yi Road Hangzhou 310036, P.R. China

Colloquium Mathematicum, Tome 96 (2003) no. 2, pp. 225-234

Résumé

Let $M$ be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n+p}(1)$. By using the Sobolev inequalities of P. Li to get $L_p$ estimates for the norms of certain tensors related to the second fundamental form of $M$, we prove some rigidity theorems. Denote by $H$ and $\| \sigma \| _p$ the mean curvature and the $L_p$ norm of the square length of the second fundamental form of $M$. We show that there is a constant $C$ such that if $\| \sigma \| _{n/2} C,$ then $M$ is a minimal submanifold in the sphere $S^{n+p-1}(1+H^2)$ with sectional curvature $ 1+H^2. $

Détail

DOI : 10.4064/cm96-2-7

Keywords: compact submanifold parallel mean curvature vector embedded unit sphere using sobolev inequalities get estimates norms certain tensors related second fundamental form prove rigidity theorems denote sigma mean curvature norm square length second fundamental form there constant sigma minimal submanifold sphere p sectional curvature