On constant higher order mean curvature hypersurfaces in Hn x R

We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H-n x 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 H-n x R: we show that compact connected hypersurfaces of constant r-th mean curvature embedded in H-n x [0, infinity) with boundary in the slice H-n x {0} are topological disks under suitable assumptions.