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

TL;DR

This paper proves a half-space theorem for constant mean curvature 1/2 entire graphs in a specific 3D space, showing that properly immersed surfaces on the mean convex side are vertical translates of the original graph.

Contribution

It establishes a new half-space theorem for CMC 1/2 graphs in (-1,) space, extending classical results to this geometric setting.

Findings

Properly immersed CMC 1/2 surfaces on the mean convex side are vertical translates of the entire graph.

The theorem applies to both mean convex and non-mean convex sides of the graph.

Provides a characterization of surfaces lying on one side of a CMC 1/2 graph in (-1,).

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 $\Sigma$.