Mean curvature one surfaces in hyperbolic space, and their relationship to minimal surfaces in Euclidean space

TL;DR

This paper explores the similarities and differences between minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space, including methods to solve their global period problems and recent developments.

Contribution

It provides a comprehensive comparison of these surfaces and introduces techniques for solving their global period problems, supported by visual examples.

Findings

Identified local similarities and global differences between the two types of surfaces.

Developed methods to solve global period problems for CMC-1 surfaces in hyperbolic space.

Presented recent results and computer graphics of example surfaces.

Abstract

We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean curvature 1 surfaces in hyperbolic 3-space, and we give an overview of recent results on these surfaces. We include computer graphics of a number of examples.