On the stability of constant higher mean curvature hypersurfaces in a Riemannian manifold

TL;DR

This paper introduces a new stability concept for constant k-mean curvature hypersurfaces in Riemannian manifolds, linking stability to eigenvalues and demonstrating instability of certain rotational spheres.

Contribution

It defines a generalized stability notion for these hypersurfaces, relates it to eigenvalues, and applies it to show instability of specific rotational spheres.

Findings

Stability relates to first eigenvalues of new operators

Embedded rotational spheres in Hnx R or Snx R are unstable

The stability notion coincides with classical one in space forms

Abstract

We propose a notion of stability for constant k-mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. Our approach led us to work with two different stability operators and we are able to relate stability to the study of the respective first eigenvalues. Moreover, we prove that embedded rotational spheres with constant k-mean curvature in Hnx R or in SnxR are not stable.