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

In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type I× fMn, where Mn is a connected n-dimensional oriented Riemannian manifold and f: I→ 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 Mn and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity. © 2017, Unione Matematica Italiana.