Local rigidity of inversive distance circle packing

Guo, Ren

doi:10.1090/S0002-9947-2011-05239-6

Local rigidity of inversive distance circle packing
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by Ren Guo

Trans. Amer. Math. Soc. 363 (2011), 4757-4776

DOI: https://doi.org/10.1090/S0002-9947-2011-05239-6

Published electronically: April 11, 2011

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Abstract:

A Euclidean (or hyperbolic) circle packing on a triangulated closed surface with prescribed inversive distance is locally determined by its cone angles. We prove this by establishing a variational principle.

References

E. M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 81 (123) (1970), 445–478 (Russian). MR 0259734
E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
Walter Brägger, Kreispackungen und Triangulierungen, Enseign. Math. (2) 38 (1992), no. 3-4, 201–217 (German). MR 1189006
Philip L. Bowers and Monica K. Hurdal, Planar conformal mappings of piecewise flat surfaces, Visualization and mathematics III, Math. Vis., Springer, Berlin, 2003, pp. 3–34. MR 2046999
Philip L. Bowers and Kenneth Stephenson, Uniformizing dessins and Belyĭ maps via circle packing, Mem. Amer. Math. Soc. 170 (2004), no. 805, xii+97. MR 2053391, DOI 10.1090/memo/0805
Alexander I. Bobenko and Boris A. Springborn, Variational principles for circle patterns and Koebe’s theorem, Trans. Amer. Math. Soc. 356 (2004), no. 2, 659–689. MR 2022715, DOI 10.1090/S0002-9947-03-03239-2
H. S. M. Coxeter, Inversive distance, Ann. Mat. Pura Appl. (4) 71 (1966), 73–83. MR 203568, DOI 10.1007/BF02413734
Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655–669 (French). MR 1106755, DOI 10.1007/BF01245096
Bennett Chow and Feng Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003), no. 1, 97–129. MR 2015261
Junfei Dai, Xianfeng David Gu, and Feng Luo, Variational principles for discrete surfaces, Advanced Lectures in Mathematics (ALM), vol. 4, International Press, Somerville, MA; Higher Education Press, Beijing, 2008. [Author name on title page: Junei Dai]. MR 2439807
D. Glickenstein, Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds. arXiv:0906.1560, to appear in J. Differential Geom.
Ren Guo, A note on circle patterns on surfaces, Geom. Dedicata 125 (2007), 175–190. MR 2322547, DOI 10.1007/s10711-007-9150-z
Ren Guo and Feng Luo, Rigidity of polyhedral surfaces. II, Geom. Topol. 13 (2009), no. 3, 1265–1312. MR 2496046, DOI 10.2140/gt.2009.13.1265
Zheng-Xu He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1–33. MR 1680531, DOI 10.2307/121018
M. K. Hurdal, K. Stephenson, Discrete conformal methods for cortical brain flattening. Neuroimage, vol. 45, S86-S98, 2009.
P. Koebe, Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88 (1936), 141–164.
Gregory Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002), 361–391. MR 1914573, DOI 10.2140/gt.2002.6.361
F. Luo, Rigidity of polyhedral surfaces. arXiv:math.GT/0612714, to appear in J. Differential Geom.
Al Marden and Burt Rodin, On Thurston’s formulation and proof of Andreev’s theorem, Computational methods and function theory (Valparaíso, 1989) Lecture Notes in Math., vol. 1435, Springer, Berlin, 1990, pp. 103–115. MR 1071766, DOI 10.1007/BFb0087901
Igor Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. (2) 139 (1994), no. 3, 553–580. MR 1283870, DOI 10.2307/2118572
Boris A. Springborn, A variational principle for weighted Delaunay triangulations and hyperideal polyhedra, J. Differential Geom. 78 (2008), no. 2, 333–367. MR 2394026
Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, Cambridge, 2005. The theory of discrete analytic functions. MR 2131318
W. P. Thurston, Geometry and topology of $3$-manifolds. lecture notes, Math. Dept. of Princeton University, 1979.