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 [4] 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 [9] 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 [11], verifying a conjecture in [11]. As a consequence, we show that the discrete Laplacian operator determines a spherical polyhedral metric.