The half space property for cmc $1/2$ graphs in $\mathbb{E}(-1,τ)$

Abstract: In this paper, we prove a half-space theorem with respect to constant mean curvature $1/2$ entire graphs in $\mathbb{E(-1,\tau)}$. If $\Sigma$ is such an entire graph and $\Sigma'$ is a properly immersed constant mean curvature $1/2$ surface included in the mean convex side of $\Sigma$ then $\Sigma'$ is a vertical translate of $\Sigma$. We also have an equivalent statement for the non mean convex side of $\Sigma$.