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 $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ 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 $H=\frac{1}{2}$, the graph is entire.