Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus

TL;DR

This paper introduces a method to construct a family of complete constant mean curvature 1 surfaces in hyperbolic space from minimal surfaces in Euclidean space, highlighting their irreducibility and symmetry properties.

Contribution

It provides a novel construction technique linking minimal surfaces in Euclidean space to CMC-1 surfaces in hyperbolic space, including irreducibility results and parameter estimates for genus 0 cases.

Findings

Constructed one-parameter family of CMC-1 surfaces in hyperbolic space.

Proved irreducibility of the constructed surfaces.

Provided parameter range estimates for genus 0 surfaces.

Abstract

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces. Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family.