Variational formulas of higher order mean curvatures

Details Variational formulas of higher order mean curvatures Variational formulas of higher order mean curvatures

2011-11-11

arxiv.org/abs/1111.2650v3

Mathematics

Differential Geometry

13 pages, to appear in SCIENCE CHINA Mathematics 2012

Scientific paper

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $\mathcal {M}_{2p}$, 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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