Willmore Submanifolds in a Sphere

Let x : M → Sn+p be an n-dimensional submanifold in an (n + p)dimensional unit sphere Sn+p, x : M → Sn+p is called a Willmore submanifold if it is an extremal submanifold to the following Willmore functional: ∫ M (S − nH) 2 dv, where S = ∑ α,i,j (hij) 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 Sn+1 and gave a characterization of Willmore tori. In this paper, we generalize this result to n-dimensional compact Willmore submanifolds in Sn+p. In fact, we obtain an integral inequality of Simons’ type for compact Willmore submanifolds in Sn+p and give a characterization of Willmore tori and Veronese surface by use of our integral inequality.