Geometry & Topology

Rigidity of polyhedral surfaces, III

Feng Luo

Full-text: Open access

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.

Article information

Source
Geom. Topol., Volume 15, Number 4 (2011), 2299-2319.

Dates
Received: 17 January 2011
Revised: 27 August 2011
Accepted: 27 September 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732368

Digital Object Identifier
doi:10.2140/gt.2011.15.2299

Mathematical Reviews number (MathSciNet)
MR2862158

Zentralblatt MATH identifier
1242.52027

Subjects
Primary: 14E20: Coverings [See also 14H30] 54C40: Algebraic properties of function spaces [See also 46J10]
Secondary: 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15} 20C20: Modular representations and characters

Keywords
polyhedral surface curvature rigidity circle packing discrete curvature

Citation

Luo, Feng. Rigidity of polyhedral surfaces, III. Geom. Topol. 15 (2011), no. 4, 2299--2319. doi:10.2140/gt.2011.15.2299. https://projecteuclid.org/euclid.gt/1513732368


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