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

Abstract

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.