On constant higher order mean curvature hypersurfaces in ${\mathbb{H}}^n\times \mathbb{R}$

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