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Abstract

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. $