Mathematical Research Letters

Volume 28 (2021)

Number 4

A new proof of Bowers–Stephenson conjecture

Pages: 1283 – 1306

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n4.a15

Author

Xu Xu (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Abstract

Inversive distance circle packing on surfaces was introduced by Bowers–Stephenson [7] 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 [19] and Luo [25] respectively. The author [34] 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 proof in [19, 25, 34] and could be generalized to three dimensional case. The new proof also reveals more properties of the inversive distance circle packing on surfaces.

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The research of the author is supported by Hubei Provincial Natural Science Foundation of China under grant no. 2017CFB681, Fundamental Research Funds for the Central Universities and National Natural Science Foundation of China under grant no. 61772379 and no. 11301402.

Received 30 December 2019

Accepted 25 May 2020

Published 22 November 2021