Properly immersed minimal surfaces in a slab of H(2)xR, H(2) the hyperbolic plane

TL;DR

This paper proves that properly immersed minimal surfaces in a bounded slab of H(2)×R have multi-graph ends, and embedded, simply connected surfaces are entire graphs, advancing understanding of minimal surface geometry in hyperbolic spaces.

Contribution

It establishes new geometric properties of minimal surfaces in H(2)×R, specifically regarding the structure of their ends within a slab of height less than π.

Findings

Ends are multi-graphs in the slab

Embedded surfaces have graph ends

Simply connected embedded surfaces are entire graphs

Abstract

We prove that the ends of a properly immersed simply or one connected minimal surface in H(2)xR contained in a slab of height less than \pi of H(2)xR, are multi-graphs. When such a surface is embedded then the ends are graphs. When embedded and simply connected, it is an entire graph.