Random Delaunay triangulations, the Thurston-Andreev theorem, and metric uniformization

TL;DR

This thesis establishes a link between discrete and continuous conformal geometry by proving a disk pattern theorem and introducing an energy functional to measure metric uniformity on Riemannian surfaces.

Contribution

It introduces a novel energy functional based on triangulation angle data and applies it to average over Delaunay triangulations for metric uniformization.

Findings

Proved a disk pattern production theorem using an angle-based energy.

Defined an energy measuring metric uniformity on Riemannian surfaces.

Connected discrete triangulation properties with continuous conformal geometry.

Abstract

In this thesis a connection between the worlds of discrete and continuous conformal geometry is explored. Specifically, a disk pattern production theroem is proved using an energy which measures how ``uniform'' the angle data of a triangulation is, see also math.DG/0002150. Then this energy is averaged over all the Delaunay triangulation of a Riemannian surface to form an energy measuring how ``uniform'' a metric is, see also math.DG/0010316.