A new proof of Bowers-Stephenson conjecture

TL;DR

This paper provides a new, simplified variational proof of the Bowers-Stephenson conjecture on the rigidity of inversive distance circle packings on surfaces, extending previous results to a broader range of inversive distances.

Contribution

The paper introduces a novel variational approach to prove the Bowers-Stephenson conjecture for inversive distances in (-1,+∞), simplifying existing proofs and enabling potential generalization to three dimensions.

Findings

Established global rigidity for inversive distances in (-1,+∞)

Provided a simplified proof method compared to previous approaches

Potentially extended the proof technique to three-dimensional cases

Abstract

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the global rigidity of circle packing with inversive distance in $(-1,+\infty)$. In this paper, we give a new variational proof of the Bowers-Stephenson conjecture for inversive distance in $(-1,+\infty)$, which simplifies the existing proofs and could be generalized to three dimensional case.