Hypersurfaces with $${H_{r+1}}$$ = 0 in $${\mathbb{H}^n \times \mathbb{R}}$$Hn×R

We prove the existence of rotational hypersurfaces in \({\mathbb{H}^n \times \mathbb{R}}\) with \({H_{r+1} = 0}\) (r-minimal hupersurfaces) and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type theorem for two ended complete hypersurfaces.