Constant mean curvature surfaces with two ends in hyperbolic space

TL;DR

This paper explores the differences between minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space, revealing new non-revolutionary examples in hyperbolic space through rigorous verification of computer experiments.

Contribution

It demonstrates the existence of genus 1 catenoid cousins in hyperbolic space, which are complete immersed surfaces with two ends that are not surfaces of revolution.

Findings

Existence of genus 1 catenoid cousins in hyperbolic space.

Complete immersed CMC 1 surfaces with two ends that are not surfaces of revolution.

Rigorous verification of computer experiments confirming these surfaces.

Abstract

We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are well-understood surfaces of revolution -- the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfaces in Euclidean 3-space, there do exist complete connected immersed constant mean curvature 1 surfaces with two ends in hyperbolic space that are not surfaces of revolution -- the genus 1 catenoid cousins. The genus 1 catenoid cousins are of interest because they show that, although minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space are intimately related, there are essential differences between these two sets of surfaces (when the surfaces are considered globally). The proof we give of existence of the genus 1 catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.