The Slab Theorem for Minimal Surfaces in $\mathbb{E}(-1,\tau)$

Abstract

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,\tau)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,\tau)$ is such a graph. More specifically: we introduce the definition of a generalized slab in $\mathbb{E}(-1,\tau)$ and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.