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

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.