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.
Highlights
In recent years, the study of complete hypersurfaces in Riemannian manifolds has attracted many geometers
We will study the uniqueness for complete hypersurfaces in weighted Riemannian warped products M n+1
Let ψ : Σn ⟶ M n+1 be a complete hypersurface with nonvanishing f -mean curvature which lies in a slab
Summary
The study of complete hypersurfaces in Riemannian manifolds has attracted many geometers. In [2], Wei and Wylie studied the complete n-dimensional weighted Riemannian manifold and proved the weighted mean curvature and volume comparison results under the ∞-Bakry-Émery Ricci tensor is bounded from below and f or ∣∇f ∣ is bounded. By using the weak maximum principle, we provide the sign relationship among the f -mean curvature and the derivative of the warping function. These auxiliary results will be the key to obtaining our results. We establish the uniqueness results for complete hypersurfaces under appropriate conditions on the f -mean curvature and the warping function in weighted Riemannian warped products M n+1 = I × ρMnf whose fiber Mnf has f -parabolic universal covering.
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