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

Abstract

In a weighted Killing warped productMn f ×ρR with warping metric 〈 , 〉M+ 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 weighted mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schrödinger 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. 2010 Mathematics Subject Classification: Primary: 58J55, 35B32, 53C42; Secondary: 35P15.