Complete Flat Surfaces with two Isolated Singularities in Hyperbolic 3-space

TL;DR

This paper constructs and classifies complete flat surfaces in hyperbolic 3-space with specific singularity and end conditions, advancing understanding of their geometric properties.

Contribution

It provides explicit examples and a classification of complete embedded flat surfaces with limited singularities in hyperbolic 3-space.

Findings

Examples of flat surfaces over a two-punctured horosphere

Classification of complete embedded flat surfaces with one end and up to two singularities

Insights into the structure of flat surfaces in hyperbolic space

Abstract

We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.