Circle packings and hyperbolic surfaces of finite type

TL;DR

This paper develops a unified method using circle packings to construct hyperbolic polyhedral metrics on surfaces of various topological types, incorporating curvature parameters and combinatorial tools.

Contribution

It introduces a novel approach that unifies the construction of hyperbolic polyhedral metrics through curvature parameters and combinatorial total geodesic curvature.

Findings

Established existence and uniqueness of packings

Unified approach for different curvature types

Extended to broader topological surface types

Abstract

This paper constructs hyperbolic polyhedral metrics via circle packings. We introduce the curvature of circles as a parameter to include all three types of constant curvature curves in the hyperbolic geometry. This provides a unified approach to producing polyhedral metrics for surfaces of broader topological types. The combinatorial total geodesic curvature serves as an effective tool for establishing the existence and uniqueness of the packing.