Abstract
We extend a technique due to Romero et al. (Class Quantum Gav 30:1–13, 2013) establishing sufficient conditions to guarantee the parabolicity of a complete two-sided hypersurface immersed into a Killing warped product $$M\times _{\rho }{\mathbb {R}}$$ , whose base $$M^n$$ has parabolic universal Riemannian covering. Afterwards, we apply our parabolicity criterium in order to obtain a rigidity result concerning these hypersurfaces, whose mean curvature is not supposed a priori be constant. Finally, parametric uniqueness results are applied to obtain suitable non-parametric ones, i.e., to the case of entire Killing graphs.
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