CMC-1 trinoids in hyperbolic 3-space and metrics of constant curvature one with conical singularities on the 2-sphere

TL;DR

This paper provides a comprehensive classification of CMC-1 trinoids in hyperbolic 3-space, including both irreducible and reducible cases, and explores their geometric properties and metrics with conical singularities.

Contribution

It offers the first explicit description of reducible CMC-1 trinoids in hyperbolic 3-space, extending previous classifications to include these cases.

Findings

Complete classification of irreducible CMC-1 trinoids

Explicit description of reducible CMC-1 trinoids

Analysis of metrics with conical singularities on the 2-sphere

Abstract

CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 trinoids in H^3 that includes the reducible case.