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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Rigidity of polyhedral surfaces, III

Feng Luo

Geometry & Topology 15 (2011) 2299–2319
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.

To Dennis Sullivan on the occasion of his seventieth birthday

Keywords
polyhedral surface, curvature, rigidity, circle packing, discrete curvature
Mathematical Subject Classification 2010
Primary: 14E20, 54C40
Secondary: 46E25, 20C20
References
Publication
Received: 17 January 2011
Revised: 27 August 2011
Accepted: 27 September 2011
Published: 3 December 2011
Proposed: David Gabai
Seconded: Dmitri Burago, Jean-Pierre Otal
Authors
Feng Luo
Department of Mathematics
Rutgers University
New Brunswick NJ 08854
USA