Curvature and rigidity of Willmore submanifolds

Abstract

Let $M$ be an $n$-dimensional compact Willmore submanifold in an $(n + p)$-dimensional unit sphere $S^{n+p}$. Denote by $S$ and $H$ the square of the length of the second fundamental form and the mean curvature of $M$. Let $\rho$ be the non-negative function on $M$ defined by $\rho^{2} = S - nH^{2}$ and $K$ be the function which assigns to each point of $M$ the infimum of the sectional curvature at the point. In this paper, first of all, we prove that, if $K$, $H$ and $p$ satisfy $K \geq \frac{p-1}{2p-1} + (n - 2)\frac{H\rho}{\sqrt{n(n-1)}}+H^{2}$, then either $M$ is totally umbilic; or a Willmore torus $W_{1,n- l}$; or the Veronese surface in $S^4$; if the Ricci curvature $R_{ii}$, $H$ and $\rho$ satisfy $R_{ii} \geq (n - 2)+(n - 2)H{\rho} + H^{2}$, for $n \geq 5$, then either $M$ is totally umbilic or a Willmore torus $W_{m,m}$. Secondly, we consider the Willmore submanifold with flat normal connection, we obtain that, if $0 \leq \rho^{2} \leq n$ then eigher $M$ is totally umbilic or a Willmore torus $W_{m,n-m}$; if $K \geq (n-2)+ \frac{H\rho}{\sqrt{n(n-1)}}+ H^{2}$, then $M$ is totally umbilic or $n \leq \rho^{2} \leq np$.