Slab theorem and halfspace theorem for constant mean curvature surfaces in H2 x R

Abstract

We prove that a properly embedded annular end of a surface in H 2 ×R with constant mean curvature 0 < H≤ 1 / 2 0<H≤1/2 can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature 0 < H ≤ 1 / 2 0<H≤1/2 contained in H2 × [ 0 , + ∞ ) H 2 ×[0,+∞) and with finite topology is necessarily a graph over a simply connected domain of H 2 H 2 . For the case H = 1 / 2 H=1/2, the graph is entire.