Rigidity of Polyhedral Surfaces, III

By Feng Luo. 2010

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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 [2] 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 [9] 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 [12], verifying a conjecture in [12]. As a consequence, we show that the discrete Laplacian operator determines a Euclidean polyhedral metric up to scaling.

Subfields: Applied Mathematics, Computational Mechanics

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