Discrete Riemann Surfaces

Investigating the Facets of Discrete Complex Analysis

Riemann surfaces arise in complex analysis as the natural domain of holomorphic functions. They are oriented two-dimensional real manifolds with a conformal structure. Several discretizations of Riemann surfaces exist, e.g., involving discretized Cauchy-Riemann equations, patterns of circles, or discrete conformal equivalence of triangle meshes. Project A01 aims at developing a comprehensive theory including discrete versions of theorems such as uniformization, convergence issues and connections to mathematical physics.


We plan to develop a comprehensive theory of discrete Riemann surfaces. The aim is to discretize the notions and theorems of complex analysis.

Scientific Details+

The theory of discrete Riemann surfaces that we envisage should be comprehensive in different respects:

Not one, but several sensible definitions of discrete holomorphic functions are known today. The oldest approach is to discretize the Cauchy-Riemann equations, and this leads to a linear theory. Other definitions, leading to nonlinear theories, originate from ideas of Thurston and involve patterns of circles. The most recent discrete model of a Riemann surface is based on a discretized notion of conformal equivalence for triangulated surfaces. We want to clarify the relationship between these linear and nonlinear theories and develop a unified theory.

The discrete theory should ideally be as rich and well developed as the classical smooth theory. We will focus on proving discrete versions of the Riemann mapping theorem and classical uniformization theorems. As it turns out, nonlinear theories of conformal maps are closely related to the theory of polyhedra in hyperbolic 3-space. Uniformization theorems for discrete Riemann surfaces are equivalent to realization theorems for hyperbolic polyhedra with prescribed dihedral angels or prescribed intrinsic metric.

As an ultimate goal, the discrete theory should contain the classical smooth theory as a limiting case. We will investigate the convergence of discrete conformal maps to smooth ones in different situations, as the discretization is refined. The aim is twofold. On the theoretical side, the discrete theory should become a source of new proofs for theorems belonging to the smooth theory. On the practical side, due to the abundance of convex variational principles, the discrete theory lends itself to numerical computation. If one could prove convergence, the discrete theory would lead to versatile new numerical methods to solve problems of conformal mapping and in Riemann surface theory.

A theory of discrete conformal maps should be accompanied by a compatible theory of discrete quasiconformal maps. There should be a notion of discrete quasiconformal distortion, which is zero only for discrete conformal maps. When considering mapping problems which are not solvable in the class of conformal maps (like mapping between conformally inequivalent Riemann surfaces, or mapping between planar domains with prescribed boundary values) one may ask for maps with least distortion. On the theoretical side one would hope for helpful characterizations of such optimal quasiconformal maps. The holy grail in this strand of research would be a discrete version of the classical Teichmüller theorem. On the practical side, one can hope for reasonable algorithms to compute optimal quasiconformal maps.

As a preliminary exercise for proving the geometrization conjecture using Ricci flow on 3-dimensional Riemannian manifolds, Richard Hamilton provided proofs of the classical uniformization theorem using Ricci flow on surfaces. Several discretized versions of Ricci flow for triangulated surfaces have been proposed, but they all suffer from the wrong scaling behavior. We will study a "correct" Ricci flow for triangulated surfaces.

We are interested in the development of a discrete theory of conformality not only for its own sake, but also because of the connections to other areas of mathematics:

Discrete conformal models in statistical physics and quantum field theory. Many 2-dimensional discrete models of statistical physics exhibit conformally invariant properties in the thermodynamic limit. This has been proved in different cases by Smirnov and Kenyon, and in each case the linear theory of discrete holomorphic functions has been instrumental. Bazhanov and others have connected the nonlinear theory of circle patterns with an important model in quantum field theory. We plan to investigate the role that the nonlinear theories of conformality play in statistical physics. In particular, there seems to be a connection between the new discrete theory of conformally equivalent metrics and the dimer model.


  • Niklas C Affolter.
    Möbius invariant Y-systems (cluster structures) for Miquel dynamics.
    preprint, December 2023.
  • Alexander I. Bobenko and Carl O. R. Lutz.
    Decorated discrete conformal equivalence in non-Euclidean geometries.
    preprint, October 2023.
  • Alexander I. Bobenko and Carl O. R. Lutz.
    Decorated discrete conformal maps and convex polyhedral cusps.
    preprint, May 2023.
  • Carl O. R. Lutz.
    Canonical tessellations of decorated hyperbolic surfaces.
    Geom Dedicata, 217(14):1–37, April 2023.
    arXiv:2206.13461, doi:10.1007/s10711-022-00746-y.
  • Hana Dal Poz Kouřimská.
    Discrete Yamabe Problem for Polyhedral Surfaces.
    Discrete & Computational Geometry, March 2023.
    arXiv:2103.15693, doi:10.1007/s00454-023-00484-2.
  • Niklas Christoph Affolter, Béatrice de Tilière, and Paul Melotti.
    The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions.
    Combinatoral Theory, July 2022.
    URL: https://escholarship.org/uc/item/2jq67049, arXiv:2208.00239.
  • Niklas Christoph Affolter, Béatrice de Tilière, and Paul Melotti.
    The Schwarzian octahedron recurrence (dSKP equation) II: geometric systems.
    preprint, July 2022.
  • Niklas Affolter, Terrence George, and Sanjay Ramassamy.
    Cross-ratio dynamics and the dimer cluster integrable system.
    preprint, August 2021.
  • Mark Gillespie, Boris Springborn, and Keenan Crane.
    Discrete Conformal Equivalence of Polyhedral Surfaces.
    ACM Trans. Graph., 40(4):103:1–103:20, August 2021.
  • Ulrich Pinkall and Boris Springborn.
    A discrete version of Liouville's theorem on conformal maps.
    Geom Dedicata, 214:389–398, April 2021.
    arXiv:1911.00966, doi:10.1007/s10711-021-00621-2.
  • Boris Springborn.
    Ideal Hyperbolic Polyhedra and Discrete Uniformization.
    Discrete Comput. Geom., September 2019.
    arXiv:1707.06848, doi:10.1007/s00454-019-00132-8.
  • Niklas Affolter, Max Glick, Pavlo Pylyavskyy, and Sanjay Ramassamy.
    Vector-relation configurations and plabic graphs.
    Selecta Mathematica, August 2019.
    URL: https://link.springer.com/article/10.1007/s00029-023-00898-z, arXiv:1908.06959.
  • Hana Kouřimská and Boris Springborn.
    Discrete Yamabe problem for polyhedral surfaces.
    Preprint, 2019.
  • U. Bücking.
    On rigidity and convergence of circle patterns.
    Discrete Comput. Geom., 61(2):380–420, 2019.
  • Alexander I Bobenko and Ulrike Bücking.
    Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere.
    Preprint at arXiv, September 2018.
  • Ulrike Bücking.
    Conformally symmetric triangular lattices and discrete θ-conformal maps.
    preprint, August 2018.
  • Niklas C Affolter.
    Miquel Dynamics, Clifford Lattices and the Dimer Model.
    Letters in Mathematical Physics, August 2018.
    URL: https://link.springer.com/article/10.1007/s11005-021-01406-0, arXiv:1808.04227.
  • Ulrike Bücking.
    $C^\infty $-convergence of conformal mappings for conformally equivalent triangular lattices.
    Results in Mathematics, 73(2):84, June 2018.
    arXiv:1706.09145, doi:10.1007/s00025-018-0845-2.
  • Alexander I Bobenko and Ananth Sridhar.
    Abelian Higgs vortices and discrete conformal maps.
    Letters in Mathematical Physics, 108(2):249–260, 2018.
    arXiv:1703.04735, doi:10.1007/s11005-017-1004-5.
  • Alexander I. Bobenko and Pascal Romon.
    Discrete CMC surfaces in $R^3$ and discrete minimal surfaces in $S^3$: a discrete Lawson correspondence.
    Journal of Integrable Systems, 2(1):1–18, May 2017.
    URL: https://academic.oup.com/integrablesystems/article/2/1/xyx010/4344752, arXiv:1705.01053.
  • Felix Günther, Caigui Jiang, and Helmut Pottmann.
    Smooth polyhedral surfaces.
    preprint, March 2017.
  • S. Sechelmann A.I. Bobenko, U. Bücking.
    Discrete minimal surfaces of Koebe type.
    In R. Verge-Rebelo D. Levi and P. Winternitz, editors, Modern Approaches to Discrete Curvature, pages 259–291. Springer, 2017.
  • Alexander I Bobenko and Felix Günther.
    Discrete Riemann surfaces based on quadrilateral cellular decompositions.
    Advances in Mathematics, 311:885–932, 2017.
    arXiv:1511.00652, doi:10.1016/j.aim.2017.03.010.
  • Alexander I Bobenko, Nikolay Dimitrov, and Stefan Sechelmann.
    Discrete Uniformization of Polyhedral Surfaces with Non-positive Curvature and Branched Covers over the Sphere via Hyper-ideal Circle Patterns.
    Discrete & Computational Geometry, 57(2):431–469, 2017.
  • U. Bücking.
    Introduction to linear and nonlinear integrable theories in discrete complex analysis.
    In R. Verge-Rebelo D. Levi and P. Winternitz, editors, Symmetries and Integrability of Difference Equations: Lecture Notes of the Abecederian of SIDE 12, Montréal 2016, CRM Ser. Math. Phys., pages 153–193. Springer, 2017.
  • Stefan Born, Ulrike Bücking, and Boris Springborn.
    Quasiconformal Dilatation of Projective Transformations and Discrete Conformal Maps.
    Discrete & Computational Geometry, 57(2):305–317, 2017.
  • B. Springborn.
    The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms.
    Enseign. Math., 63(3-4):333–373, 2017.
  • H. Kourimska, L. Skuppin, and B. Springborn.
    A variational principle for cyclic polygons with prescribed edge lengths.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • U. Bücking.
    Approximation of conformal mappings using confomally equivalent triangular lattices.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • U. Bücking and D. Matthes.
    Constructing solutions to the Björling problem for isothermic surfaces by structure preserving discretization.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • A. I. Bobenko and F. Günther.
    Discrete complex analysis on planar quad-graphs.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • A. I. Bobenko, S. Sechelmann, and B. Springborn.
    Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • F. Bornemann, A. Its, S. Olver, and G. Wechslberger.
    Numerical Methods for the Discrete Map $Z^a$.
    In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
  • A. I. Bobenko, N. Dimitrov, and S. Sechelmann.
    Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns.
    preprint, 2015.
  • N. Dimitrov.
    Hyper-ideal Circle Patterns with Cone Singularities.
    Results in Mathematics, pages 1–45, 2015.
    arXiv:1406.6741, doi:10.1007/s00025-015-0453-3.
  • A. I. Bobenko and A. Its.
    The asymptotic behaviour of the discrete holomorphic map $Z^a$ via the Riemann-Hilbert method.
    Duke Math. J., 2015. accepted.
  • Alexander Bobenko and Mikhail Skopenkov.
    Discrete Riemann surfaces: linear discretization and its convergence.
    J. reine und angew. Math., October 2014.
    arXiv:1210.0561, doi:10.1515/crelle-2014-0065.
  • A.I. Bobenko and B. Springborn.
    Diskretisierung in Geometrie und Dynamik - Elastische Stäbe und Rauchringe.
    Mitteilungen der DMV, 21(1):218–224, December 2013.
    URL: http://www.degruyter.com/view/j/dmvm.2013.21.issue-00004/issue-files/dmvm.2013.21.issue-00004.xml.
  • Ivan Izmestiev, Robert B. Kusner, Günter Rote, Boris Springborn, and John M. Sullivan.
    There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems.
    Geometriae Dedicata, 166(1):15–29, October 2013.
    arXiv:1207.3605, doi:10.1007/s10711-012-9782-5.
  • Alexander I. Bobenko and Felix Günther.
    Discrete complex analysis – the medial graph approach.
    Actes des rencontres du CIRM 3 no. 1: Courbure discrète: théorie et applications, pages 159–169, 2013.
    URL: http://acirm.cedram.org/acirm-bin/fitem?id=ACIRM_2013__3_1_159_0, doi:10.5802/acirm.65.

PhD thesis


Prof. Dr. Alexander I. Bobenko   +

Projects: A01, A02, B02, C01, CaP, Z
University: TU Berlin, Institut für Mathematik, MA 881
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424655
E-Mail: bobenko[at]math.tu-berlin.de
Website: http://page.math.tu-berlin.de/~bobenko/

Dr. Ulrike Bücking   +

Projects: A01
University: FU Berlin, Institut für Mathematik, 038
Address: Arnimallee 3, 14195 Berlin, GERMANY
Tel: +49 30 83866595
E-Mail: buecking[at]math.fu-berlin.de
Website: https://www.mi.fu-berlin.de/en/math/groups/ag-diskret-algebra-geom/members/Professoren/ulrike_buecking.html

Prof. Dr. Boris Springborn   +

Projects: A01
University: TU Berlin, Institut für Mathematik, MA 871
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31423617
Fax: +49 30 31479282
E-Mail: boris.springborn[at]tu-berlin.de
Website: http://page.math.tu-berlin.de/~springb/

Dr. Niklas Affolter   +

Projects: A01
University: TU Berlin, Institut für Mathematik, MA 861
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31479252
E-Mail: affolter[at]math.tu-berlin.de
Website: https://page.math.tu-berlin.de/~affolter/

Fabian Bittl   +

Projects: A01
University: TU Berlin, Institut für Mathematik, MA 863
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31422232
E-Mail: fabian.bittl[at]tu-berlin.de
Website: https://page.math.tu-berlin.de/~bittl/

Carl O. R. Lutz   +

Projects: A01
University: TU Berlin, Institut für Mathematik, MA 883
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31429426
E-Mail: clutz[at]math.tu-berlin.de
Website: https://page.math.tu-berlin.de/~clutz/