Rigidity of minimal submanifolds in space forms

TL;DR

This paper establishes a rigidity theorem for certain minimal submanifolds in space forms, showing they must be totally umbilical spheres under specific integral Ricci curvature bounds, extending previous pointwise results.

Contribution

It extends earlier rigidity results from pointwise Ricci curvature bounds to integral Ricci curvature bounds for minimal submanifolds with parallel mean curvature.

Findings

Submanifolds are totally umbilical spheres under the given curvature conditions.

The result generalizes previous theorems to integral Ricci curvature bounds.

Provides explicit bounds depending on geometric parameters.

Abstract

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