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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 someuniqueness theorems for $r$-minimal hypersurfaces with a given (finite orasymptotic) boundary. In particular, we obtain a Schoen-type Theorem for twoended complete hypersurfaces.

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