Slab theorem and halfspace theorem for constant mean curvature surfaces in ${\mathbb{H}}^2\times \mathbb{R}$

Abstract

We prove that a properly embedded annular end of a surface in ${\mathbb{H}}^2\times \mathbb{R}$ with constant mean curvature $0<H\le 1/2$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\le 1/2$ contained in ${\mathbb{H}}^2\times [0,+\infty )$ and with finite topology is necessarily a graph over a simply connected domain of ${\mathbb{H}}^2$. For the case $H=1/2$, the graph is entire.