Rigidity of Polyhedral Surfaces, III

TL;DR

This paper proves the rigidity of inversive distance circle packings and shows that polyhedral metrics are uniquely determined by discrete curvatures, advancing understanding of polyhedral surface rigidity.

Contribution

It confirms the Bowers-Stephenson conjecture on the rigidity of inversive distance circle packings and links polyhedral metrics to discrete curvatures for the first time.

Findings

Proved the rigidity conjecture for inversive distance circle packings.

Demonstrated that polyhedral metrics are determined by discrete curvatures.

Showed the discrete Laplacian operator determines a polyhedral metric up to scaling.

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 as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive distance circle packings are rigid. Using a recent work of R. Guo on variational principle associated to the inversive distance circle packing, we prove rigidity conjecture of Bowers-Stephenson in this paper. We also show that each polyhedral metric on a triangulated surface is determined by various discrete curvatures introduced in our previous work, verifying a conjecture in \cite{Lu1}. As a consequence, we show that the discrete Laplacian operator determines a Euclidean polyhedral metric up to scaling.