Rigidity of polyhedral surfaces, III

Abstract

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P Bowers and K Stephenson in [Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc. (2004)] as a generalization of Andreev and Thurston’s circle packing. They conjectured that inversive distance circle packings are rigid. We prove this conjecture using recent work of R Guo [Trans. Amer. Math. Soc. 363 (2011) 4757–4776] on the variational principle associated to the inversive distance circle packing. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures that we introduced in [arXiv 0612.5714], verifying a conjecture in [arXiv 0612.5714]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.