Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products

Abstract

In a Riemannian warped product I×fMn, where I⊂R is an open interval, f is a positive real function defined on I and Mn is a compact Riemannian manifold without boundary, we use equivariant bifurcation theory in order to establish sufficient conditions, in terms of f and the spectrum of the Laplacian on Mn, that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are H2-hypersurfaces, namely, whose boundaries are hypersurfaces with constant second mean curvature H2. For each of our results, we have provided a considerable number of examples that verify all the assumptions under consideration.