Constructions of $H_r$-hypersurfaces, barriers and Alexandrov Theorem in $H^n \times R$

Abstract: In this paper, we are concerned with hypersurfaces in $H^n\times R$ with constant r-mean curvature, to be called $H_r$-hypersurfaces. We construct examples of complete $H_r$-hypersurfaces which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $H_r$-graph for each value $0<H_r\leq\frac{n-r}{n}$. Also, for each value $H_r>\frac{n-r}{n}$, there is a unique embedded compact strictly convex rotational $H_r$-hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an Alexandrov-type Theorem. Namely, we prove that an embedded compact $H_r$-hypersurface in $H^n\times R$ is rotational ($H_r>0$).