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Discrete Riemann surfaces: Linear discretization and its convergence

Alexander Bobenko and Mikhail Skopenkov


We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann–Roch theorem. The proofs use energy estimates inspired by electrical networks.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: Collaborative Research Center SFB/TR 109 Discretization in Geometry and Dynamics

Funding statement: The first author was partially supported by the DFG Collaborative Research Center SFB/TR 109 “Discretization in Geometry and Dynamics”. The second author was partially supported by the President of the Russian Federation grant MK-5490.2014.1, by “Dynasty” foundation, and by the Simons–IUM fellowship. Part of the work on this paper was done during the stay of the second author at King Abdullah University of Science and Technology in Saudi Arabia.


The authors are grateful to D. Chelkak, S. von Deylen, I. Dynnikov, A. Gaifullin, F. Günther, S. Lando, C. Mercat, A. Pakharev, M. Wardetzky for useful discussions and to S. Tikhomirov for writing a software for numerical experiments.


[1] Baker M. and Norine S., Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math. 215 (2007), no. 2, 766–788. 10.1016/j.aim.2007.04.012Search in Google Scholar

[2] Bobenko A. and Springborn B., A discrete Laplace–Beltrami operator for simplicial surfaces, Discrete Comput. Geom. 38 (2007), 740–756. 10.1007/s00454-007-9006-1Search in Google Scholar

[3] Bobenko A. I., Introduction to compact Riemann surfaces, Computational approach to Riemann surfaces, Lecture Notes in Math. 2013, Springer, Berlin (2011), 3–64. 10.1007/978-3-642-17413-1_1Search in Google Scholar

[4] Bobenko A. I., Mercat C. and Schmies M., Period matrices of polyhedral surfaces, Computational approach to Riemann surfaces, Lecture Notes in Math. 2013, Springer, Berlin (2011), 213–226. 10.1007/978-3-642-17413-1_7Search in Google Scholar

[5] Bobenko A. I., Mercat C. and Suris Y. B., Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function, J. reine angew. Math. 583 (2005), 117–161. 10.1515/crll.2005.2005.583.117Search in Google Scholar

[6] Bobenko A. I., Pinkall U. and Springborn B., Discrete conformal maps and ideal hyperbolic polyhedra, preprint 2010, 10.2140/gt.2015.19.2155Search in Google Scholar

[7] Bohle C., Pedit F. and Pinkall U., Discrete holomorphic geometry I. Darboux transformations and spectral curves, J. reine angew. Math. 637 (2009), 99–139. 10.1515/CRELLE.2009.092Search in Google Scholar

[8] Bücking U., Approximation of conformal mappings by circle patterns, Geom. Dedicata 137 (2008), 163–197. 10.1007/s10711-008-9292-7Search in Google Scholar

[9] Chelkak D. and Smirnov S., Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math. 189 (2012), 515–580. 10.1007/s00222-011-0371-2Search in Google Scholar

[10] Chelkak D. and Smirnov S., Discrete complex analysis on isoradial graphs, Adv. Math. 228 (2011), no. 3, 1590–1630. 10.1016/j.aim.2011.06.025Search in Google Scholar

[11] Courant R., Friedrichs K. and Lewy H., Über die partiellen Differentialgleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32–74. 10.1007/BF01448839Search in Google Scholar

[12] Duffin R. J., Discrete potential theory, Duke Math. J. 20 (1953), 233–251. 10.1215/S0012-7094-53-02023-7Search in Google Scholar

[13] Duffin R. J., Distributed and lumped networks, J. Math. Mech. 8 (1959), 793–826. 10.1512/iumj.1959.8.58051Search in Google Scholar

[14] Dynnikov I. A. and Novikov S. P., Geometry of the triangle equation on two-manifolds, Mosc. Math. J. 3 (2003), no. 2, 419–438. 10.17323/1609-4514-2003-3-2-419-438Search in Google Scholar

[15] Ferrand J., Fonctions préharmoniques et fonctions préholomorphes, Bull. Sci. Math. (2) 68 (1944), 152–180. Search in Google Scholar

[16] He Z.-X. and Schramm O., On the convergence of circle packings to the Riemann map, Invent. Math. 125 (1996), no. 2, 285–305. 10.1007/s002220050076Search in Google Scholar

[17] Isaacs R., A finite difference function theory, Univ. Nac. Tucuman. Revista A. 2 (1941), 177–201. Search in Google Scholar

[18] Kenyon R., Conformal invariance of domino tiling, Ann. Probab. 28 (2000), no. 2, 759–795. 10.1214/aop/1019160260Search in Google Scholar

[19] Lelong-Ferrand J., Représentation conforme et transformations à intégrale de Dirichlet bornée, Gauthier-Villars, Paris 1955. Search in Google Scholar

[20] Mercat C., Discrete Riemann surfaces and the Ising model, Comm. Math. Phys. 218 (2001), no. 1, 177–216. 10.1007/s002200000348Search in Google Scholar

[21] Mercat C., Discrete period matrices and related topics, preprint 2002, Search in Google Scholar

[22] Mercat C., Discrete polynomials and discrete holomorphic approximation, preprint 2002, Search in Google Scholar

[23] Mercat C., Discrete Riemann surfaces, Handbook of Teichmüller theory, vol. I, IRMA Lect. Math. Theor. Phys. 11, European Mathematical Society, Zürich (2007), 541–575. 10.4171/029-1/14Search in Google Scholar

[24] Meyer M., Desbrun M., Schröder P. and Barr A. H., Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and mathematics III, Springer, Berlin (2003), 35–57. 10.1007/978-3-662-05105-4_2Search in Google Scholar

[25] Pakharev A., Skopenkov M. and Ustinov A., Through the resisting net, Mat. Prosv. 18 (2014), 33–65. Search in Google Scholar

[26] Pinkall U. and Polthier K., Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993), no. 1, 15–36. 10.1080/10586458.1993.10504266Search in Google Scholar

[27] Rivin I., Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. (2) 139 (1994), no. 3, 553–580. 10.2307/2118572Search in Google Scholar

[28] Rodin B. and Sullivan D., The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. 10.4310/jdg/1214441375Search in Google Scholar

[29] Schramm O., Circle patterns with the combinatorics of the square grid, Duke Math. J. 86 (1997), 347–389. 10.1007/978-1-4419-9675-6_8Search in Google Scholar

[30] Skopenkov M., The boundary value problem for discrete analytic functions, Adv. Math. 240 (2013), 61–87. 10.1016/j.aim.2013.03.002Search in Google Scholar

[31] Skopenkov M., Prasolov M. and Dorichenko S., Dissections of a metal rectangle, Kvant 3 (2011), 10–16. Search in Google Scholar

[32] Skopenkov M., Smykalov V. and Ustinov A., Random walks and electric networks, Mat. Prosv. 16 (2012), 25–47. Search in Google Scholar

[33] Smirnov S., Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits, C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244. 10.1016/S0764-4442(01)01991-7Search in Google Scholar

[34] Smirnov S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. of Math. (2) 172 (2010), no. 2, 1435–1467. 10.4007/annals.2010.172.1435Search in Google Scholar

[35] Springborn B., Schröder P. and Pinkall U., Conformal equivalence of triangle meshes, ACM SIGGRAPH 2008, ACM, New York (2008), article no. 77. Search in Google Scholar

[36] Stephenson K., Introduction to circle packing. The theory of discrete analytic functions, Cambridge University Press, Cambridge 2005. Search in Google Scholar

[37] Thurston W. P., The geometry and topology of three-manifolds, lecture notes 2002, Search in Google Scholar

[38] Troyanov M., Les surface euclidiennes à singularités coniques, Enseign. Math. 32 (1986), 79–94. Search in Google Scholar

[39] Wilson S., Conformal cochains, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5247–5264. 10.1090/S0002-9947-08-04556-XSearch in Google Scholar

Received: 2013-11-26
Revised: 2014-4-25
Published Online: 2014-8-19
Published in Print: 2016-11-1

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