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 $0\<H\leq 1/2$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0\<H\leq 1/2$ 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.