Variational formulas of higher order mean curvatures

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional of a submanifold M (n) in a general Riemannian manifold N (n+m) for . As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional , called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms, as well as a special variational problem.