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

We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$withconstantmeancurvature$0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\le \frac{1}{2}$ contained in ${\mathbb{H}}^2\times [0,+\mathrm{\infty })$ 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.