Uniqueness of hypersurfaces in weighted product spaces via maximum principles for the drift Laplacian

We apply suitable maximum principles for the drift Laplacian to obtain several uniqueness results concerning complete two-sided hypersurfaces immersed with constant f -mean curvature in a weighted product space of form R × M f n and such that its potential function f does not depend on the parameter t ∈ R . Among these results, we prove that the slices are the only complete two-sided f -minimal hypersurfaces lying in a half-space of R × M f n and such that the Bakry–Émeri–Ricci tensor is bounded from below. Furthermore, we study the f -mean curvature equation related to entire graphs defined on the base M n .