Rigidity of submanifolds with parallel mean curvature in space froms

TL;DR

This paper classifies compact submanifolds with parallel mean curvature in space forms, showing they are either spheres, Clifford hypersurfaces, or complex projective spaces under certain Ricci curvature conditions.

Contribution

It provides a rigidity theorem characterizing submanifolds with parallel mean curvature based on Ricci curvature bounds, extending previous classifications.

Findings

Submanifolds are either totally umbilic spheres, Clifford hypersurfaces, or complex projective spaces.

A strict Ricci curvature inequality implies the submanifold is a totally umbilic sphere.

The classification applies to submanifolds in simply connected space forms with positive curvature.

Abstract

Let $M$ be an $n(\geq3)$-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 $Ric_{M}\geq(n-2)(c+H^2),$ then $M$ is either a totally umbilic sphere, the Clifford hypersurface $S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)$ 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 $Ric_{M}>(n-2)(c+H^2),$ then $M$ is a totally umbilic sphere.