Combinatorial curvature flows for generalized hyperbolic circle packings

TL;DR

This paper introduces new combinatorial curvature flows for generalized hyperbolic circle packings, providing algorithms to achieve prescribed curvatures and analyzing their long-term behaviors.

Contribution

It proposes novel combinatorial Calabi flows and their variants for generalized hyperbolic circle packings, extending previous work and offering effective computational methods.

Findings

Established equivalence conditions for flow behaviors

Provided algorithms for prescribed curvature packings

Analyzed long-term convergence of the flows

Abstract

Generalized circle packings were introduced in \cite{Ba-Hu-Sun} as a generalization of tangential circle packings in hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow, fractional combinatorial Calabi flow and combinatorial $p$-th Calabi flow for generalized hyperbolic circle packings. We establish several equivalent conditions regarding the longtime behaviors of these flows. This provides effective algorithms for finding the generalized circle packings with prescribed total geodesic curvatures.