Global pinching theorems for minimal submanifolds in spheres

Volume 96 / 2003

Kairen Cai Colloquium Mathematicum 96 (2003), 225-234 MSC: Primary 53C20, 53C40. DOI: 10.4064/cm96-2-7


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. $


  • Kairen CaiDepartment of Mathematics
    Hangzhou Teachers' College
    96 Wen Yi Road
    Hangzhou 310036, P.R. China

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