Willmore Submanifolds in a sphere

By Haizhong Li. 2002

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Let x : M → S n+p be an n-dimensional submanifold in an (n + p)- dimensional unit sphere S n+p , x : M → S n+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: � M (S − nH 2 ) n 2 dv, where S = � α,i,j (h α ) 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 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 Willmore tori and Veronese surface by use of our integral inequality.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ffe62ee96e9b99e2d786728082c7318e2d6ac27f