Rigidity of polyhedral surfaces, III

Luo, Feng¹

¹ Department of Mathematics, Rutgers University, New Brunswick NJ 08854, USA

Geometry & topology, Tome 15 (2011) no. 4, pp. 2299-2319.

Résumé

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.

Détail

DOI : 10.2140/gt.2011.15.2299

Classification : 14E20, 54C40, 46E25, 20C20
Keywords: polyhedral surface, curvature, rigidity, circle packing, discrete curvature

Bibliographie

[1] E M Andreev, Convex polyhedra in Lobachevskii spaces, Mat. Sb. (N.S.) 81 (123) (1970) 445

[2] A Bobenko, U Pinkall, B Springborn, Discrete conformal maps and ideal hyperbolic polyhedra

[3] P L Bowers, M K Hurdal, Planar conformal mappings of piecewise flat surfaces, from: "Visualization and mathematics III" (editors H C Hege, K Polthier), Math. Vis., Springer (2003) 3

[4] P L Bowers, K Stephenson, Uniformizing dessins and Belyĭmaps via circle packing, Mem. Amer. Math. Soc. 170, no. 805, Amer. Math. Soc. (2004)

[5] B Chow, F Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003) 97

[6] H Cohn, R Kenyon, J Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14 (2001) 297

[7] D Glickenstein, Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds, J. Differential Geom. 87 (2011) 201

[8] X D Gu, R Guo, F Luo, W Zeng, Discrete Laplace–Beltrami operator determines discrete Riemannian metric

[9] R Guo, Local rigidity of inversive distance circle packing, Trans. Amer. Math. Soc. 363 (2011) 4757

[10] G Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002) 361

[11] F Luo, Rigidity of polyhedral surfaces, to appear in J. Differential Geom.

[12] F Luo, A characterization of spherical polyhedral surfaces, J. Differential Geom. 74 (2006) 407

[13] F Luo, On Teichmüller spaces of surfaces with boundary, Duke Math. J. 139 (2007) 463

[14] I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994) 553

[15] K Stephenson, Introduction to circle packing: The theory of discrete analytic functions, Cambridge Univ. Press (2005)

[16] W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979)

[17] Y Colin De Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991) 655