A CLASS OF VARIATIONAL PROBLEMS FOR SUBMANIFOLDS IN A SPACE FORM

Abstract. Let Rn+p (c) be an (n + p)-dimensional space form of constant sectional curvature c, x: M → Rn+p (c) an n-dimensional submanifold in Rn+p (c). For k with k ≥ 1, x: M → Rn+p (c) is called an extremal submanifold if it is a critical submanifold to the following non-negative functional: Fk(x): = (S − nH M 2) k dv, where S = ∑ (h α,i,j α ij)2 is the square of the length of the second fundamental form, and H is the mean curvature of M. We note that when k = n, the 2 above functional is Willmore functional. In this paper, we prove an integral inequality of Simons ’ type for n-dimensional compact extremal submanifolds in the (n + p)-dimensional unit sphere Sn+p and give a characterization of Clifford torus and Veronese surface by use of our integral inequality. Case k = 1 and case k = n were studied by Guo-Li [4] and Li [12]. 2 1