Convergences of Combinatorial Ricci Flows to Degenerated Circle Packings in Hyperbolic Background Geometry

TL;DR

This paper studies degenerated circle packings in hyperbolic geometry, providing conditions for realization, proving uniqueness, and introducing a Ricci flow method to find such packings.

Contribution

It characterizes conditions for realizing degenerated circle packings and introduces a Ricci flow approach in hyperbolic geometry.

Findings

Characterized necessary and sufficient conditions for degenerated circle packings.

Proved the uniqueness of the circle packings under these conditions.

Developed a Ricci flow method to construct the packings.

Abstract

This paper investigates a kind of degenerated circle packings in hyperbolic background geometry. A main problem is whether a prescribed total geodesic curvature data can be realized by a degenerated circle packing or not. We fully characterize the sufficient and necessary conditions and show the uniqueness. Furthermore, we introduce the combinatoral Ricci flow to find the desired degenerated circle packed surface, analougus to the methods of Chow-Luo and Takatsu.