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Global pinching theorems for minimal submanifolds in spheres

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Cai K.

Colloquium Mathematicum

2003

tom 96

nr 2

225-234

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 $||σ||_{p}$ 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 $||σ||_{n/2} < C$, then M is a minimal submanifold in the sphere $S^{n+p-1}(1+H²)$ with sectional curvature 1+H².

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1 Colloquium Mathematicum

1 Cai K.

1 2003

Wyniki wyszukiwania

Global pinching theorems for minimal submanifolds in spheres