Combinatorial Calabi flow for ideal circle pattern

TL;DR

The paper develops and analyzes the combinatorial Calabi flow for ideal circle patterns on closed surfaces in both hyperbolic and Euclidean backgrounds. It proves global existence for all time and exponential convergence of the flow to an ideal circle pattern metric realizing a given attainable curvature vector $\overline{K}$, by leveraging a strictly convex potential whose Hessian equals the curvature Jacobian. This yields a practical algorithm to compute ideal circle patterns via iterative radius updates and connects Thurston’s curvature map to discrete geometric flows and ideal hyperbolic polyhedra. The results extend prior work on combinatorial flows by establishing convergence in both geometries and providing a rigorous foundation for pattern construction.

Abstract

We study the combinatorial Calabi flow for ideal circle patterns in both hyperbolic and Euclidean background geometry. We prove that the flow exists for all time and converges exponentially fast to an ideal circle pattern metric on surfaces with prescribed attainable curvatures. As a consequence, we provide an algorithm to find the desired ideal circle patterns.

Figures (2)

• Figure 1: An ideal circle pattern.
• Figure 2: A two-circle configuration.

Theorems & Definitions (19)

• Definition 1
• Theorem A: Bobenko-Springborn BS
• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7