Combinatorial Calabi flow for ideal circle pattern

TL;DR

This paper introduces a combinatorial Calabi flow method that guarantees exponential convergence to ideal circle pattern metrics on surfaces, providing an effective algorithm for constructing such patterns in hyperbolic and Euclidean geometries.

Contribution

The paper establishes the global existence and exponential convergence of the combinatorial Calabi flow for ideal circle patterns, offering a new algorithmic approach.

Findings

Flow exists for all time and converges exponentially.

Algorithm effectively finds ideal circle patterns with prescribed curvatures.

Works in both hyperbolic and Euclidean geometries.

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.