Rigidity of minimal submanifolds in space forms

Abstract

In this paper, we consider the rigidity for an $n(\ge 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb{M}}_c^{n+p}$ when the integral Ricci curvature of $M$ has some bound. We prove that, if $c+H^2>0$ and $\Vert {\mathrm{Ric}}_{-}^{\lambda }{\Vert }_{n/2}<\epsilon (n,c,\lambda ,H)$ for $\lambda$ satisfying $\frac{n-2}{n-1}(c+H^2)<\lambda \le c+H^2$, then $M$ is the totally umbilical sphere ${\mathbb{S}}^n(\frac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel mean curvature of $M$, and $\epsilon (n,c,\lambda ,H)$ 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.

9 pages