On the Stability and Parabolicity of Complete f-Minimal Hypersurfaces in Weighted Warped Products

Abstract

Our purpose in this paper is to study the stability and parabolicity of f-minimal hypersurfaces immersed in a weighted warped product of the type $$I\times _{\rho }M^n_f$$ , where $$M^n_f$$ stands for a weighted manifold endowed with a weight function f. In this setting, we obtain sufficient conditions to guarantee that an f-minimal hypersurfaces immersed in $$I\times _{\rho }M^n_f$$ be $$L_f$$ -stable, where $$L_f$$ stands for the weighted Jacobi operator. Moreover, we establish a criterion of f-parabolicity and we apply it to infer when an f-minimal hypersurface immersed in $$I\times _{\rho }M^n_f$$ is totally geodesic. A study of the existence of f-minimal entire graphs in $$I\times _{\rho }M^n_f$$ is also made.