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

Abstract

In a weighted Killing warped product $M^n_f\times_\rho\mathbb R$ with war\-ping me\-tric $\langle\,,\rangle_{\!M}+{\rho}^2\,dt$, where the warping function $\rho$ is a real positive function defined on $M^n$ and the weighted function $f$ does not depend on the parameter $t\in\mathbb R$, 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 $\{\Omega_{\gamma}\}_{\gamma\in I}$ whose boundaries $\partial\Omega_{\gamma}$ 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 $x\colon \Sigma^n\hookrightarrow M^n_f\times_\rho\mathbb R$ with constant weighted mean curvature in terms of the first eigenvalue of the $f$-Laplacian of $\Sigma^n$.