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arXiv · 1511.03170

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

Vanderson Lima
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Abstract

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,τ)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,τ)$ is such a graph. More specifically: we introduce the definition of a generalized slab in $\mathbb{E}(-1,τ)$ 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.

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Vanderson Lima. 2017-06-21. The Slab Theorem for Minimal Surfaces in $\mathbb{E}(-1,τ)$. https://doi.org/10.1007/s10455-016-9531-3

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