Mathematical Research Letters
Volume 28 (2021)
A new proof of Bowers–Stephenson conjecture
Pages: 1283 – 1306
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 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.
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