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.

Final version. Typos and minor errors corrected. The counter-example presented in the first version was excluded due to a problem

References in corpus (1)

• Non-properly Embedded Minimal Planes in Hyperbolic 3-Space

Cited by in corpus (2)

• On the asymptotic Plateau problem in ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$
• On the asymptotic Plateau problem for area minimizing surfaces in $\mathbb{E}(-1,\tau )$