On 2k-Minimal Submanifolds

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 functional.

In this paper, we examine the critical points of the total 2k-th Gauss–Bonnet curvature functional, called 2k-minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)-mean curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to 2k-minimal submanifolds.