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.
To Dennis Sullivan on the occasion of
his seventieth birthday
Keywords
polyhedral surface, curvature, rigidity, circle packing,
discrete curvature
