The ideal Thurston-Andreev theorem and triangulation production

TL;DR

This paper generalizes the Thurston-Andreev theorem for higher genus surfaces, introducing a conformal flow of angle data that converges to a unique hyperbolic structure, linked to hyperbolic volume.

Contribution

It extends the ideal Thurston-Andreev theorem to higher genus surfaces and develops a conformal flow method for angle data convergence.

Findings

The flow converges to a unique uniform angle data.

The objective function is related to hyperbolic volume.

Provides a new approach to triangulation production for complex surfaces.

Abstract

This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be the gradient of an objective function related in a rather magical way to hyperbolic volume.