Hypersurfaces with H_{r+1}=0 in H x R

We prove the existence of rotational hypersurfaces in $$\mathbb{H}$^n\times $\mathbb{R}$$with$H_{r+1}=0$ 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.