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| In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional of a n-dimensional submanifold M in a general (n+m)-dimensional Riemannian manifold N. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the total 2p-th mean curvature 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. |
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| Keywords:Differential Geometry; 2p-minimal; mean curvature; austere submanifold |
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