On $(2k)$-Minimal Submanifolds

TL;DR

This paper introduces and characterizes $(2k)$-minimal submanifolds as critical points of the total $(2k)$-th Gauss-Bonnet curvature, generalizing classical minimal submanifold properties.

Contribution

It defines $(2k)$-minimal submanifolds via higher curvature functionals and establishes their characterization through the vanishing of a higher mean curvature.

Findings

$(2k)$-minimal submanifolds are critical points of the total $(2k)$-th Gauss-Bonnet curvature.

They are characterized by the vanishing of the $(2k+1)$-Gauss-Bonnet curvature.

Several properties of classical minimal submanifolds extend to $(2k)$-minimal cases.

Abstract

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the $(2k+1)$-Gauss-Bonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to $(2k)$-minimal submanifolds.