On the deformation of inversive distance circle packings, II

TL;DR

This paper extends previous results on inversive distance circle packings to hyperbolic geometry, proving long-term existence and convergence of the combinatorial Ricci flow, and exploring curvature ranges.

Contribution

It generalizes earlier theorems to hyperbolic settings and analyzes the behavior of the Ricci flow and curvature ranges in this context.

Findings

Ricci flow solutions exist for all time in hyperbolic geometry

Solutions converge exponentially fast under zero curvature conditions

Generalizes Andreev-Thurston's theorem to hyperbolic background geometry

Abstract

We show that the results in \cite{Ge-Jiang1} are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a metric with zero curvature. We also give some results about the range of discrete Gaussian curvatures, which generalize Andreev-Thurston's theorem to some extent.