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Transactions of the American Mathematical Society

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Variational principles for circle patterns and Koebe's theorem

Authors: Alexander I. Bobenko and Boris A. Springborn
Journal: Trans. Amer. Math. Soc. 356 (2004), 659-689
MSC (2000): Primary 52C26; Secondary 53A30
Published electronically: September 22, 2003
MathSciNet review: 2022715
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Abstract: We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdière's, Brägger's and Rivin's functionals can be derived from ours.

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  • [Bow91] B. H. Bowditch, Singular Euclidean structures on surfaces, J. London Math. Soc. (2) 44 (1991), no. 3, 553-565. MR 93i:57014
  • [Brä92] W. Brägger, Kreispackungen und Triangulierungen, Enseign. Math. 38 (1992), 201-217. MR 94b:52032
  • [BS93] G. R. Brightwell and E. R. Scheinerman, Representations of planar graphs, SIAM J. Discrete Math. 6 (1993), no. 2, 214-229. MR 95d:05043
  • [BS02] A. I. Bobenko and Yu. B. Suris, Integrable systems on quad-graphs, Internat. Math. Res. Notices 2002, no. 11, 573-612. MR 2003d:37127
  • [CdV91] Y. Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), 655-669. MR 92h:57020
  • [DS95] T. Dubejko and K. Stephenson, Circle packing: experiments in discrete analytic function theory, Experiment. Math. 4 (1995), no. 4, 307-348. MR 97f:57027
  • [FF62] L. R. Ford, Jr. and D. R. Fulkerson, Flows in networks, Princeton University Press, Princeton, NJ, 1962. MR 28:2917
  • [Gar92] B. T. Garrett, Circle packings and polyhedral surfaces, Discrete Comput. Geom. 8 (1992), 429-440. MR 93g:52014
  • [Gib77] P. J. Giblin, Graphs, surfaces and homology, Chapman and Hall, London, 1977. MR 55:11235
  • [HBS$^{+}$99] M. K. Hurdal, P. L. Bowers, K. Stephenson, De Witt L. Sumners, K. Rehm, K. Schaper, and D. A. Rottenberg, Quasi-conformally flat mapping the human cerebellum, Medical Image Computing and Computer-Assisted Intervention--MICCAI '99 (Berlin) (Ch. Taylor and A. Colchester, eds.), Lecture Notes in Computer Science, vol. 1679, Springer-Verlag, 1999, pp. 279-286.
  • [Koe36] P. Koebe, Kontaktprobleme der konformen Abbildung, Abh. Sächs. Akad. Wiss. Leipzig Math.-Natur. Kl. 88 (1936), 141-164.
  • [Lei01] G. Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002), 363-391. MR 2003c:52034
  • [Lew81] L. Lewin, Polylogarithms and associated functions, North Holland, New York, 1981. MR 83b:33019
  • [Mar01] D. Martindale, Road map for the mind, Scientific American 285 (2001), 13.
  • [Mer01] Ch. Mercat, Discrete Riemann surfaces and the Ising model, Commun. Math. Phys. 218 (2001), 177-216. MR 2002c:82019
  • [Moh93] B. Mohar, A polynomial time circle packing algorithm, Discrete Math. 117 (1993), 257-263. MR 94h:52038
  • [Riv94] I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994), 553-580. MR 96h:57010
  • [Riv96] -, A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. 143 (1996), 51-70. MR 96i:52008
  • [Riv99] -, Combinatorial optimization in geometry, Preprint arXiv:math.GT/9907032, July 1999, To appear in Adv. in Appl. Math.
  • [Sac94] H. Sachs, Coin graphs, polyhedra, and conformal mappings, Discrete Math. 134 (1994), 133-138. MR 95j:52020
  • [Sch92] O. Schramm, How to cage an egg, Invent. Math. 107 (1992), no. 3, 543-560. MR 93c:52009
  • [Sch02] J. M. Schlenker, Hyperbolic manifolds with polyhedral boundary, Preprint. ArXiv:math.GT/0111136, v.5, September 2002.
  • [SR34] E. Steinitz and H. Rademacher, Vorlesungen über die Theorie der Polyeder, Springer-Verlag, Berlin, 1934. MR 55:3962 (reprint)
  • [Ste22] E. Steinitz, Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, vol. 3 (Geometrie), 1922, Part 3AB12, pp. 1-139.
  • [Thu] W. P. Thurston, The geometry and topology of three-manifolds, electronic version 1.0 of 1997. A version is currently available from the Mathematical Sciences Research Institute at the URL
  • [Zie95] G. M. Ziegler, Lectures on polytopes, Springer-Verlag, 1995. MR 96a:52011

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Additional Information

Alexander I. Bobenko
Affiliation: Institut für Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

Boris A. Springborn
Affiliation: Institut für Mathematik, MA 8-5, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

Received by editor(s): July 23, 2002
Published electronically: September 22, 2003
Additional Notes: The research was partially supported by the Sonderforschungsbereich 288
Article copyright: © Copyright 2003 American Mathematical Society

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