Variational formulas of higher order mean curvatures

Abstract

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.

13 pages, to appear in SCIENCE CHINA Mathematics 2012

References in corpus (2)

• A proof of the DDVV conjecture and its equality case
• A survey on the DDVV conjecture

Cited by in corpus (1)

• On $f$-bi-harmonic maps between Riemannian manifolds