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 × ρ 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.

New Bernstein Type Results in Weighted Warped Products

“…In Section 2, we introduce some basic notions and facts to be used for hypersurfaces immersed in weighted warped products. In Section 3, we prove some parametric results related to the hypersurfaces in a weighted warped product that will enable us to obtain our main uniqueness result ( eorem 1) which extend the corresponding results in [13]. To conclude this paper, we will devote Section 4 to prove a new Bernstein type result for entire graphs in weighted warped products for the constant weighted mean curvature case ( eorem 2).…”

“…In the following, we give the next technical lemma (for further details, see Proposition 3.3 in [13]) which will be essential for the proofs of our main results. Moreover, we also need to consider the weighted warped products I× ρ M n f satisfy the following convergence condition:…”

“…Furthermore, de Lima et al [12] obtained Liouville type results for two-sided hypersurfaces in weighted Killing warped products. More recently, the author [13] proved some uniqueness results of complete hypersurfaces in weighted Riemannian warped products I× ρ M n f with f-parabolic fiber, through the application of the weak maximum principle.…”