Willmore submanifolds in a sphere

Abstract

Let $x:M\to S^{n+p}$ be an $n$-dimensional submanifold in an $(n+p)$-dimensional unit sphere $S^{n+p}$, $x:M\to S^{n+p}$ is called a Willmore submanifold to the following Willmore functional: $${\int }_M(S-n{H}^2{)}^{\frac{n}{2}}dv,$$ where $S=\sum _{\alpha ,i,j}(h_{ij}^{\alpha }{)}^2$ is the square of the length of the second fundamental form, $H$ is the mean curvature of $M$. In [13], author proved an integral inequality of Simon's type for $n$-dimensional compact Willmore hypersurfaces in $S^{n+1}$ and gave a characterization of {\it Willmore tori}. In this paper, we generalize this result to $n$-dimensional compact Willmore submanifolds in $S^{n+p}$. In fact, we obtain an integral inequality of Simon's type for compact Willmore submanifolds in $S^{n+p}$ and give a characterization of {\it willmore tori} and {\it Veronese surface} by use of integral inequality.

18 pages. To appear in Mathematical Research Letter