Parabolicity and uniqueness of complete two-sided hypersurfaces immersed in a Riemannian warped product

Abstract

In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type $$I\times _fM^n$$ , where $$M^{n}$$ is a connected n-dimensional oriented Riemannian manifold and $$f:I\rightarrow \mathbb {R}$$ is a positive smooth function. As applications, we obtain several uniqueness results concerning these hypersurfaces with constant mean curvature, under standard constraints on the Ricci curvature of $$M^n$$ and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity.