EN
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.