Mathematical Research Letters

Volume 9 (2002)

Number 6

Willmore Submanifolds in a sphere

Pages: 771 – 790

DOI: http://dx.doi.org/10.4310/MRL.2002.v9.n6.a6

Author

Haizhong Li (Tsinghua University)

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 if it is an extremal submanifold to the following Willmore functional: $$ \int_M(S-nH^2)^{\frac{n}{2}}dv, $$ where $S=\sum\limits_{\alpha,i,j}(h^\alpha_{ij})^2$ 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 Simons' 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 Simons' type for compact Willmore submanifolds in $S^{n+p}$ and give a characterization of {\it Willmore tori} and {\it Veronese surface} by use of our integral inequality.

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