A variational problem for submanifolds in a sphere

Abstract

Let $ x{:}\ M\rightarrow S^{n+p} $ be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S n + p , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional $\int_M(S-nH^2)^{\frac{n}{2}}d\nu$ , where $ S=\sum_{\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 [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S n + p . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional $\int_M(S-nH^2)d\nu$ . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.