Random Delaunay triangulations and metric uniformization

Abstract

In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately related to hyperbolic volume. With the use of random Delaunay triangulations we then average this objective function to construct an objective function on the metrics conformal to a fixed one. Finally using this averaged objective function we may reprove the uniformization theorem in two dimensions.