On the deformation of inversive distance circle packings, III

TL;DR

This paper studies the deformation of inversive distance circle packings on triangulated surfaces, proving existence, uniqueness, and convergence to constant alpha-curvature metrics using Ge-Xu's alpha-flow, and discusses obstacles to their existence.

Contribution

It extends rigidity results for circle packings with constant alpha-curvature and analyzes the long-term behavior of Ge-Xu's alpha-flow.

Findings

Uniqueness of constant alpha-curvature circle packings when alpha times Euler characteristic is non-positive.

Exponential convergence of the alpha-flow to the constant alpha-curvature metric.

Identification of combinatorial and topological obstacles to existence.

Abstract

Given a triangulated surface $M$, we use Ge-Xu's $\alpha$-flow \cite{Ge-Xu1} to deform any initial inversive distance circle packing metric to a metric with constant $\alpha$-curvature. More precisely, we prove that the inversive distance circle packing with constant $\alpha$-curvature is unique if $\alpha\chi(M)\leq 0$, which generalize Andreev-Thurston's rigidity results for circle packing with constant cone angles. We further prove that the solution to Ge-Xu's $\alpha$-flow can always be extended to a solution that exists for all time and converges exponentially fast to constant $\alpha$-curvature. Finally, we give some combinatorial and topological obstacles for the existence of constant $\alpha$-curvature metrics.