On $H = 1/2$ surfaces in $\widetilde{\mathit {PSL}}_{2}(\mathbb {R}, \tau )$

Abstract

In this paper we prove that if $\Sigma$ is a properly embedded constant mean curvature $H = 1/2$ surface which is asymptotic to a horocylinder $C \subset \widetilde{\mathit{PSL}}_{2}(\mathbb{R}, \tau)$, in one side of $C$, such that the mean curvature vector of $\Sigma$ has the same direction as that of the $C$ at points of $\Sigma$ converging to $C$, then $\Sigma$ is a subset of $C$.