Rigidity of Willmore submanifolds and extremal submanifolds in the unit sphere

Abstract

Let M be an n-dimensional $(n\ge 4)$ compact Willmore (or extremal) submanifold in the unit sphere $S^{n+p}$. Denote by ${\text {Ric}}$ and H the Ricci curvature and the mean curvature of M, respectively. It is proved that if $(\int _M ({\text {Ric}}_-^{\lambda })^\frac{n}{2})^\frac{2}{n}<A(n,\lambda ,H,\rho )~ (\text{ or }\ B(n,\lambda ,H,\rho ))$, then M is a totally umbilical sphere, where $A(n,\lambda ,H,\rho )$ and $B(n,\lambda ,H,\rho )$ are two explicit positive constants depending on n, $\lambda$, H, and $\rho$. This extends parts of the results from a pointwise Ricci curvature lower bound to an integral Ricci curvature lower bound.