Rigidity of submanifolds with parallel mean curvature in space froms

Abstract

Let $M$ be an $n(\ge 3)$-dimensional oriented compact submanifold with parallel mean curvature in the simply connected space form $F^{n+p}(c)$ with $c+H^2>0$, where $H$ is the mean curvature of $M$. We prove that if the Ricci curvature of $M$ satisfies $Ri{c}_M\ge (n-2)(c+H^2),$ then $M$ is either a totally umbilic sphere, the Clifford hypersurface $S^m\left(\frac{1}{\sqrt{2(c+H^2)}}\right)\times S^m\left(\frac{1}{\sqrt{2(c+H^2)}}\right)$ in $S^{n+1}(\frac{1}{\sqrt{c+H^2}})$ with $n=2m$, or $\mathbb{C}{P}^2(4/3(c+H^2))$ in $S^7(\frac{1}{\sqrt{c+H^2}})$. In particular, if $Ri{c}_M>(n-2)(c+H^2),$ then $M$ is a totally umbilic sphere.

11 pages

References in corpus (1)

• On Yau rigidity theorem for minimal submanifolds in spheres