Combinatorial Calabi flows with ideal circle patterns

TL;DR

This paper extends combinatorial Ricci flows to combinatorial Calabi flows for ideal circle patterns, proving long-term existence and exponential convergence to flat or hyperbolic cone metrics in both Euclidean and hyperbolic geometries.

Contribution

It introduces combinatorial Calabi flows for ideal circle patterns and proves their global existence and exponential convergence in both Euclidean and hyperbolic settings.

Findings

Solutions exist for all time

Flows converge exponentially to target metrics

Applicable in both Euclidean and hyperbolic geometries

Abstract

In this paper, we extend the work of Ge-Hua-Zhou \cite{GHZ} on combinatorial Ricci flows for ideal circle patterns to combinatorial Calabi flows in both hyperbolic and Euclidean background geometry. We prove the solution to the combinatorial Calabi flows with any given initial Euclidean (hyperbolic resp.)ideal circle pattern exists for all time and converges exponentially fast to a flat cone metric (hyperbolic resp.) on a given surface.