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

TL;DR

This paper extends the classical Slab Theorem to the homogeneous spaces (-1,), showing that properly immersed minimal surfaces with finite topology in a generalized slab are multi-graph ends, and embedded ones are graphs, with simply connected surfaces being entire graphs.

Contribution

It introduces the concept of generalized slabs in (-1,) and proves that minimal surfaces within them have multi-graph or entire graph ends, extending the classical theory to new geometric settings.

Findings

Properly immersed finite topology minimal surfaces in slabs have multi-graph ends.

Embedded minimal surfaces in slabs have graph ends.

Simply connected embedded minimal surfaces are entire graphs.

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.