Rigidity of infinite inversive distance circle packings in the plane

TL;DR

This paper proves the conjecture that infinite inversive distance circle packings in the plane are rigid, extending known results and employing new principles and lemmas related to circle packings and Delaunay triangulations.

Contribution

It establishes the rigidity of infinite inversive distance circle packings in the hexagonal plane, generalizing previous results on tangential packings.

Findings

Proved the rigidity conjecture for inversive distance circle packings in the hexagonal plane.

Developed a maximal principle for weighted Delaunay packings.

Introduced a ring lemma for inversive distance circle packings.

Abstract

In 2004, Bowers-Stephenson [2] introduced the inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured the rigidity of infinite inversive distance circle packings in the plane. Motivated by the recent work of Luo-Sun-Wu [22] on Luo's vertex scaling, we prove Bower-Stephenson's conjecture for inversive distance circle packings in the hexagonal triangulated plane. This generalizes Rodin-Sullivan's famous result [13] on the rigidity of infinite tangential circle packings in the hexagonal triangulated plane. The key tools include a maximal principle for generic weighted Delaunay inversive distance circle packings and a ring lemma for the inversive distance circle packings in the hexagonal triangulated plane.