Rigidity of minimal submanifolds in space forms

In this paper, we consider the rigidity for an $n(\ge 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}c$whentheintegralRiccicurvatureof$M$ has some bound. We prove that, if $c+H^2>0$ and $|$\mathrm{R}\mathrm{i}\mathrm{c}${-}^\lambda|_{n/2}< \epsilon(n,c, \lambda, H)$for$\lambda$satisfying$ \frac{n-2}{n-1} (c+H^2) < \lambda \le c+H^2$,then$M$isthetotallyumbilicalsphere$$\mathbb{S}$^n(\tfrac{1}{\sqrt{c+H^2}})$\mathrm{.}Here$H$ is the norm of the parallel mean curvature of $M$, and $ϵ(n,c,\lambda ,H)$ is a positive constant depending only on $n,c,\lambda$ and $H$. This extends some of the earlier work of [15] from pointwise Ricci curvature lower bound to inetgral Ricci curvature lower bound.