Abstract
We introduce the character of Thurston’s circle packings in the hyperbolic background geometry Consequently, some quite simple criteria are obtained for the existence of hyperbolic circle packings. For example, if a closed surface X admits a circle packing with all the vertex degrees di ⩾ 7, then it admits a unique complete hyperbolic metric so that the triangulation graph of the circle packing is isotopic to a geometric decomposition of X. This criterion is sharp due to the fact that any closed hyperbolic surface admits no triangulations with all di ⩽ 6. As a corollary, we obtain a new proof of the uniformization theorem for closed surfaces with genus g ⩾ 2, and moreover, any hyperbolic closed surface has a geometric decomposition. To obtain our results, we use Chow-Luo’s combinatorial Ricci flow as a fundamental tool.
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References
Andreev E. On convex polyhedra in Lobačevskiĭ spaces. Mat Sb, 1970, 81/123: 445–478
Andreev E. On convex polyhedra of finite volume in Lobačevskiĭ space. Mat Sb, 1970, 83/125: 256–260
Bao X, Bonahon F. Hyperideal polyhedra in hyperbolic 3-space. Bull Soc Math France, 2002, 130: 457–491
Bobenko A, Hoffmann T, Springborn B. Minimal surfaces from circle patterns: Geometry from combinatorics. Ann of Math (2), 2006, 164: 231–264
Bobenko A, Springborn B. Variational principles for circle patterns and Koebe’s theorem. Trans Amer Math Soc, 2004, 356: 659–689
Bowditch B. Singular Euclidean structures on surfaces. J Lond Math Soc (2), 1991, 44: 553–565
Bowers P L, Stephenson K. The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense. Math Proc Cambridge Philos Soc, 1992, 111: 487–513
Bowers P L, Stephenson K. Uniformizing Dessins and Belyč Maps via Circle Packing. Memoirs of the American Mathematical Society, vol. 805. Providence: Amer Math Soc, 2004
Chow B, Luo F. Combinatorial Ricci flows on surfaces. J Differential Geom, 2003, 63: 97–129
Dai J, Gu X, Luo F. Variational Principles for Discrete Surfaces. Beijing: Higher Ed Press, 2008
de Verdière Y C. Un principe variationnel pour les empilements de cercles. Invent Math, 1991, 104: 655–669
Edmonds A L, Ewing J H, Kulkarni R S. Regular tessellations of surfaces and (p, q, 2)-triangle groups. Ann of Math (2), 1982, 116: 113–132
Feng K, Ge H, Hua B. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds. Geom Topol, 2022, 26: 1349–1384
Ge H, Hua B. 3-dimensional combinatorial Yamabe flow in hyperbolic background geometry. Trans Amer Math Soc, 2020, 373: 5111–5140
Ge H, Hua B, Zhou Z. Combinatorial Ricci flows for ideal circle patterns. Adv Math, 2021, 383: 107698
Ge H, Hua B, Zhou Z. Circle patterns on surfaces of finite topological type. Amer J Math, 2021, 143: 1397–1430
Ge H, Jiang W. On the deformation of inversive distance circle packings, II. J Funct Anal, 2017, 272: 3573–3595
Ge H, Jiang W, Shen L. On the deformation of ball packings. Adv Math, 2022, 398: 108192
He Z X, Liu J S. On the Teichmuüller theory of circle patterns. Trans Amer Math Soc, 2013, 365: 6517–6541
He Z X, Schramm O. Fixed points, Koebe uniformization and circle packings. Ann of Math (2), 1993, 137: 369–406
He Z X, Schramm O. Rigidity of circle domains whose boundary has σ-finite linear measure. Invent Math, 1994, 115: 297–310
He Z X, Schramm O. Koebe uniformization for “almost circle domains”. Amer J Math, 1995, 117: 653–667
Hodgson C D, Rivin I. A characterization of compact convex polyhedra in hyperbolic 3-space. Invent Math, 1993, 111: 77–111
Huang X, Liu J. Characterizations of circle patterns and finite convex polyhedra in hyperbolic 3-space. Math Ann, 2017, 368: 213–231
Koebe P. Kontaktprobleme der konformen Abbildung. Ber Süachs Akad Wiss Leipzig, Math-Phys Kl, 1936, 88: 141–164
Leibon G. Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geom Topol, 2002, 6: 361–391
Liu J, Zhou Z. How many cages midscribe an egg. Invent Math, 2016, 203: 655–673
Luo F. A combinatorial curvature flow for compact 3-manifolds with boundary. Electron Res Announc Amer Math Soc, 2005, 11: 12–20
Marden A, Rodin B. On Thurston’s formulation and proof of Andreev’s theorem. In: Computational Methods and Function Theory. Lecture Notes in Mathematics, vol. 1435. Berlin: Springer, 1990, 103–115
Martelli B. An introduction to geometric topology. http://people.dm.unipi.it/martelli/geometric_topology.html, 2023
Rivin I. Euclidean structures on simplicial surfaces and hyperbolic volume. Ann of Math (2), 1994, 139: 553–580
Rivin I. A characterization of ideal polyhedra in hyperbolic 3-space. Ann of Math (2), 1996, 143: 51–70
Rodin B, Sullivan D. The convergence of circle packings to the Riemann mapping. J Differential Geom, 1987, 26: 349–360
Rousset M. Sur la rigidité de polyèdres hyperboliques en dimension 3: Cas de volume fini, cas hyperidéal, cas fuchsien. Bull Soc Math France, 2004, 132: 233–261
Schlenker J M. Hyperideal circle patterns. Math Res Lett, 2005, 12: 85–102
Schramm O. How to cage an egg. Invent Math, 1992, 107: 543–560
Stephenson K. Introduction to Circle Packing. The Theory of Discrete Analytic Functions. Cambridge: Cambridge Univ Press, 2005
Thurston W. The Geometry and Topology of Three-Manifolds. Princeton: Princeton Univ Press, 1979
Zhou Z. Circle patterns with obtuse exterior intersection angles. arXiv:1703.01768v3, 2019
Zhou Z. Producing circle patterns via configurations. arXiv:2010.13076v9, 2021
Acknowledgements
Huabin Ge was supported by National Natural Science Foundation of China (Grant Nos. 11871094 and 12122119). Aijin Lin was supported by National Natural Science Foundation of China (Grant No. 12171480), Hunan Provincial Natural Science Foundation of China (Grant Nos. 2020JJ4658 and 2022JJ10059), and Scientific Research Program Funds of National University of Defense Technology (Grant No. 22-ZZCX-016). The second author thanks Ke Feng and Ze Zhou for communication on related topics. The authors thank Liangming Shen for many useful conversations. The authors are very grateful to the referees for carefully reading the original manuscript and pointing out some typos.
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Ge, H., Lin, A. The character of Thurston’s circle packings. Sci. China Math. 67, 1623–1640 (2024). https://doi.org/10.1007/s11425-023-2182-2
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DOI: https://doi.org/10.1007/s11425-023-2182-2