On constant higher order mean curvature hypersurfaces in ℍn × ℝ

We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in ℍn × ℝ. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in ℍn × ℝ: we show that compact connected hypersurfaces of constant r-th mean curvature embedded in ℍn × [0, ∞) with boundary in the slice ℍn × {0} are topological disks under suitable assumptions.