On the deformation of inversive distance circle packings

TL;DR

This paper extends the combinatorial Ricci flow to inversive distance circle packings, enabling deformation to prescribed cone angles and demonstrating exponential convergence despite potential singularities.

Contribution

It generalizes Chow-Luo's flow to inversive distance packings and proves convergence and extendability of solutions, broadening classical geometric results.

Findings

Flow can be extended for all time despite singularities

Solutions converge exponentially fast to target packings

Partial characterization of admissible cone angles

Abstract

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flow can be used to deform any inversive distance circle packing to a unique packing with prescribed cone angle. We also give partial results on the range of all admissible cone angles, which generalize the classical Andreev-Thurston's theorem.