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 H 2 × R with constant mean curvature 0 < H ≤ 12 can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature 0 < H ≤ 12 contained in H 2 × [0 , + ∞ ) and with finite topology is necessarily a graph over a simply connected domain of H 2 . For the case H = 12 , the graph is entire. 2020 Mathematics Subject Classification: 53C30, 53A10.