Bifurcation and local rigidity of constant second mean curvature hypersurfaces in Riemannian warped products. (August 2020) - Velásquez, Marco A.L.; Ramalho, André F.A.; da Silva, Jonatan F.; Oliveira, Jobson Q.
Abstract: In a Riemannian warped product I × f M n, where I ⊂ 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.