On the classification of discrete conformal structures on surfaces

TL;DR

This paper completes the classification of discrete conformal structures on surfaces, introduces new classes including generalized circle packing metrics, and explores their connections to 3D hyperbolic geometry.

Contribution

It provides a full classification of discrete conformal structures on surfaces, extending previous work and identifying new classes and relationships.

Findings

Complete classification of discrete conformal structures

Introduction of new classes including generalized circle packing metrics

Discussion of links to 3D hyperbolic geometry

Abstract

Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification. In this paper, we give a full classification of the discrete conformal structures on surfaces, which completes Glickenstein-Thomas' classification. As a result, we find some new classes of discrete conformal structures on surfaces, including some of the generalized circle packing metrics introduced by Guo-Luo \cite{GL2}. The relationships between the discrete conformal structures on surfaces and the 3-dimensional hyperbolic geometry are also discussed.