Rigidity of polyhedral surfaces, III

Luo, Feng

doi:10.2140/gt.2011.15.2299

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Rigidity of polyhedral surfaces, III

Feng Luo

Geometry & Topology 15 (2011) 2299–2319

DOI: 10.2140/gt.2011.15.2299

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

Volume 15, issue 4 (2011)