Variational formulas of higher order mean curvatures

TL;DR

This paper derives the first variational formulas for higher order mean curvature functionals of submanifolds in Riemannian manifolds, identifying critical points and exploring their geometric properties and relations.

Contribution

It establishes the first variational formula and Euler-Lagrange equations for total $2p$-th mean curvature functionals, extending variational theory to higher order curvatures.

Findings

Closed complex submanifolds are critical points for all $p$.

Introduces the concept of relatively $2p$-minimal submanifolds.

Explores relations between relatively $2p$-minimal and austere submanifolds.

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.