Local rigidity, bifurcation, and stability of $H_f$-hypersurfaces in weighted Killing warped products

Abstract

In a weighted Killing warped product Mn f ×ρR with warping metric h , iM+ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weightedmean curvature. For this, we analyze the number of negative eigenvalues of a certain Schr¨odinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x: Σn ,→ Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f-Laplacian of Σn.