Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

Abstract

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product $I{\times }_{\rho }{M}_f^n$ whose fiber $M$ has $f$ -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.