Convergence of Discrete Conformal Maps of Surfaces

Abstract

This dissertation proves a convergence result about discrete conformalstructures on smooth manifolds. A \emph{discrete conformal structure} is a collection of triangulated piecewiseconstant-curvature manifolds where each constant-curvature piece is a simplex and each manifold in the collection has the same underlying triangulation. In this dissertation we define a discrete conformal structure on a triangulated topological manifold by assigning weights to each edge and each vertex in the triangulation. A \emph{discrete conformal map}, then, is a map between two piecewiseconstant-curvature manifolds in the same discrete conformal structure. These maps are not conformal in either the Riemannian geometry sense or the complex analysis sense, but they do share some properties with smooth conformal maps. In the literature there are several specific kinds of discrete conformalstructure that are usually studied separately. These include circle packing discrete conformal structures, circle patterns, and vertex scaling discrete conformal structures. This dissertation studies general discrete conformal structures, so the results herein hold for each of the particular examples of discrete conformal structures that are usually studied separately. Furthermore, discrete conformal maps are usually defined only on piecewiseconstant-curvature manifolds. In this dissertation, however, we use a notion of Riemannian barycentric maps to allow us to define discrete conformal maps on smooth manifolds. Our main result generalizes the Rodin-SullivanTheorem from circle packing in two senses. First, our result is valid for Riemannian surfaces without boundary, while the Rodin-Sullivan Theorem only holds for the particular case of a Jordan domain in the complex plane. Second, we consider discrete conformal structures in general so we are not constrained to a particular discrete conformal structure such as circle packing or vertex scaling discrete conformal structures.