Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Combinatorial Calabi flows on surfaces


Author: Huabin Ge
Journal: Trans. Amer. Math. Soc. 370 (2018), 1377-1391
MSC (2010): Primary 53C44, 52C26
DOI: https://doi.org/10.1090/tran/7196
Published electronically: October 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of the smooth Calabi flow. We prove that the solution to the combinatorial Calabi flow exists for all time and converges if and only if the Thurston's circle packing exists. As a consequence, the combinatorial Calabi flow provides a new algorithm to find circle packings with prescribed curvatures. The proofs rely on careful analysis of the combinatorial Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.


References [Enhancements On Off] (What's this?)

  • [1] R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664, https://doi.org/10.2307/1995057
  • [2] Eugenio Calabi, Extremal Kähler metrics, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290. MR 645743
  • [3] Eugenio Calabi, Extremal Kähler metrics. II, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 95-114. MR 780039
  • [4] Shu-Cheng Chang, The 2-dimensional Calabi flow, Nagoya Math. J. 181 (2006), 63-73. MR 2210710
  • [5] Shu-Cheng Chang, Global existence and convergence of solutions of Calabi flow on surfaces of genus $ h\geq2$, J. Math. Kyoto Univ. 40 (2000), no. 2, 363-377. MR 1787876, https://doi.org/10.1215/kjm/1250517718
  • [6] X. X. Chen, Calabi flow in Riemann surfaces revisited: a new point of view, Internat. Math. Res. Notices 6 (2001), 275-297. MR 1820328, https://doi.org/10.1155/S1073792801000149
  • [7] Bennett Chow, The Ricci flow on the $ 2$-sphere, J. Differential Geom. 33 (1991), no. 2, 325-334. MR 1094458
  • [8] Bennett Chow and Feng Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003), no. 1, 97-129. MR 2015261
  • [9] Piotr T. Chruściel, Semi-global existence and convergence of solutions of the Robinson-Trautman ($ 2$-dimensional Calabi) equation, Comm. Math. Phys. 137 (1991), no. 2, 289-313. MR 1101689
  • [10] Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
  • [11] Yves Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991), no. 3, 655-669 (French). MR 1106755, https://doi.org/10.1007/BF01245096
  • [12] Huabin Ge and Xu Xu, Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension, Adv. Math. 267 (2014), 470-497. MR 3269185, https://doi.org/10.1016/j.aim.2014.09.011
  • [13] Huabin Ge and Xu Xu, 2-dimensional combinatorial Calabi flow in hyperbolic background geometry, Differential Geom. Appl. 47 (2016), 86-98. MR 3504921, https://doi.org/10.1016/j.difgeo.2016.03.011
  • [14] David Glickenstein, A combinatorial Yamabe flow in three dimensions, Topology 44 (2005), no. 4, 791-808. MR 2136535, https://doi.org/10.1016/j.top.2005.02.001
  • [15] David Glickenstein, A maximum principle for combinatorial Yamabe flow, Topology 44 (2005), no. 4, 809-825. MR 2136536, https://doi.org/10.1016/j.top.2005.02.002
  • [16] D. Glickenstein, Geometric triangulations and discrete Laplacians on manifolds, Preprint at arXiv:math/0508188v1 [math.MG].
  • [17] David Glickenstein, A monotonicity property for weighted Delaunay triangulations, Discrete Comput. Geom. 38 (2007), no. 4, 651-664. MR 2365828, https://doi.org/10.1007/s00454-007-9009-y
  • [18] David Glickenstein, Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds, J. Differential Geom. 87 (2011), no. 2, 201-237. MR 2788656
  • [19] X. D. Gu, R. Guo, F. Luo, and W. Zeng, Discrete Laplace-Beltrami operator determines discrete Riemannian metric, Preprint at arXiv:1010.4070v1 [cs.DM].
  • [20] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306. MR 664497
  • [21] Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237-262. MR 954419, https://doi.org/10.1090/conm/071/954419
  • [22] Zheng-Xu He, Rigidity of infinite disk patterns, Ann. of Math. (2) 149 (1999), no. 1, 1-33. MR 1680531, https://doi.org/10.2307/121018
  • [23] Anil Nirmal Hirani, Discrete exterior calculus, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)-California Institute of Technology. MR 2704508
  • [24] M. Jin, J. Kim, and X. D. Gu, Discrete surface Ricci flow: theory and applications, Mathematics of Surfaces XII, Lecture Notes in Computer Science Vol. 4647, 209-232.
  • [25] Feng Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004), no. 5, 765-780. MR 2100762, https://doi.org/10.1142/S0219199704001501
  • [26] 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, https://doi.org/10.1007/BFb0087901
  • [27] Alan D. Rendall, Existence and asymptotic properties of global solutions of the Robinson-Trautman equation, Classical Quantum Gravity 5 (1988), no. 10, 1339-1347. MR 964979
  • [28] I. Robinson and A. Trautman, Spherical gravitational waves, Phys. Rev. Lett. 4 (1960), 431-432.
  • [29] Bernd G. Schmidt, Existence of solutions of the Robinson-Trautman equation and spatial infinity, Gen. Relativity Gravitation 20 (1988), no. 1, 65-70. MR 925329, https://doi.org/10.1007/BF00759256
  • [30] David Singleton, On global existence and convergence of vacuum Robinson-Trautman solutions, Classical Quantum Gravity 7 (1990), no. 8, 1333-1343. MR 1064183
  • [31] Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, Cambridge, 2005. The theory of discrete analytic functions. MR 2131318
  • [32] Michael Struwe, Curvature flows on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 2, 247-274. MR 1991140
  • [33] W. P. Thurston, Geometry and topology of 3-manifolds, Chapter 13, Princeton lecture notes 1976, http://www.msri.org/publications/books/gt3m.
  • [34] K. P. Tod, Analogues of the past horizon in the Robinson-Trautman metrics, Classical Quantum Gravity 6 (1989), no. 8, 1159-1163. MR 1005655
  • [35] W. Zeng, D. Samaras, and X. D. Gu, Ricci flow for 3D shape analysis, IEEE Transactions on Pattern Analysis & Machine Intelligence, 32(4) (2010), 662-677.
  • [36] W. Zeng, R. Shi, Z. Su, and X. D. Gu, Colon surface registration using Ricci flow, Abdomen and Thoracic Imaging, An Engineering & Clinical Perspective (2014), 389-419.
  • [37] M. Zhang, F. Li, Y. He, S. Li, D. Wang, and L. M. Lui, Registration of brainstem surfaces in adolescent idiopathic scoliosis using discrete Ricci flow, 15th International Conference, Nice, France, October 1-5, 2012, Proceedings, Part II, Medical Image Computing and Computer-Assisted Intervention CMICCAI (2012), 146-154.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44, 52C26

Retrieve articles in all journals with MSC (2010): 53C44, 52C26


Additional Information

Huabin Ge
Affiliation: Department of Mathematics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
Email: hbge@bjtu.edu.cn

DOI: https://doi.org/10.1090/tran/7196
Keywords: Circle packing, combinatorial Calabi flow, combinatorial Calabi energy, combinatorial Ricci potential
Received by editor(s): March 10, 2015
Received by editor(s) in revised form: July 5, 2016, and December 23, 2016
Published electronically: October 16, 2017
Additional Notes: This research was supported by the National Natural Science Foundation of China under grant No.11501027, and Fundamental Research Funds for the Central Universities (Nos. 2015JBM103, 2014RC028 and 2016JBM071).
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society