# Slab theorem and halfspace theorem for constant mean curvature surfaces in $H_{2}×R$

### Laurent Hauswirth

Université Gustave Eiffel; Université Paris-Est Créteil, Marne-la-Vallée, France### Ana Menezes

Princeton University, USA### Magdalena Rodríguez

Universidad de Granada, Spain

## Abstract

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

## Cite this article

Laurent Hauswirth, Ana Menezes, Magdalena Rodríguez, Slab theorem and halfspace theorem for constant mean curvature surfaces in $H_{2}×R$. Rev. Mat. Iberoam. 39 (2023), no. 1, pp. 307–320

DOI 10.4171/RMI/1372