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

In a Riemannian warped product I x (f) M-n, where I subset of R is an open interval, f is a positive real function defined on I and M-n 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 M-n, that allow us to guarantee the existence of bifurcation instants or the local rigidity of a certain family of open sets whose boundaries are H-2-hypersurfaces, namely, whose boundaries are hypersurfaces with constant second mean curvature H-2. For each of our results, we have provided a considerable number of examples that verify all the assumptions under consideration. (C) 2020 Elsevier Ltd. All rights reserved.