Properly embedded, area-minimizing surfaces in hyperbolic $3$-space

TL;DR

This paper establishes a bridge principle at infinity for area-minimizing surfaces in hyperbolic 3-space and demonstrates that any open, connected, orientable surface can be properly embedded as such, with disjoint limit sets for different ends.

Contribution

It introduces a new bridge principle at infinity and shows how to embed any open, connected, orientable surface as an area-minimizing surface in hyperbolic 3-space.

Findings

Proved a bridge principle at infinity for hyperbolic space.

Constructed proper embeddings of arbitrary open, connected, orientable surfaces.

Ensured disjoint limit sets for different ends of the embedded surfaces.

Abstract

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that the limit sets of different ends are disjoint.