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

# A01

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.
**

**Group:**A. Geometry-
**Principal Investigators:**Prof. Dr. Alexander I. Bobenko, Dr. Ulrike Bücking, Prof. Dr. Boris Springborn -
**Investigators:**Niklas Affolter, Dr. Felix Günther, Hana Kourimska, Isabella Retter, Dr. Stefan Sechelmann, Lara Skuppin, Dr. Ananth Sridhar **University:**TU Berlin**Term:**since 2012

#### Mission-

#### 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.

#### Publications+

##### Dissertations

###### Discrete Riemann surfaces and integrable systems

**Author: **
Günther, Felix

**Advisor: **Alexander I. Bobenko

**Date: **
Sep 2014

**Download: **
external

###### Spinor representation of Bryant surfaces with catenoidal and smooth ends

**Author: **
Pavljukevich, Tatiana

**Advisor: **Alexander I. Bobenko

**Date: **
2014

##### Papers

###### A variational principle for cyclic polygons with prescribed edge lengths

**Authors: **
Kourimska, H. and
Skuppin, L. and
Springborn, B.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Note: **Preprint at arxiv

**Date: **
2016

**Download: **
arXiv

###### Approximation of conformal mappings using confomally equivalent triangular lattices

**Author: **
Bücking, U.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Note: **Preprint at arxiv

**Date: **
2016

**Download: **
arXiv

###### Constructing solutions to the Björling problem for isothermic surfaces by structure preserving discretization

**Authors: **
Bücking, U. and
Matthes, D.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Note: **Preprint at arxiv

**Date: **
2016

**Download: **
arXiv

###### Discrete complex analysis on planar quad-graphs

**Authors: **
Bobenko, A. I. and
Günther, F.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Note: **Preprint at arxiv

**Date: **
2016

**Download: **
arXiv

###### Discrete conformal maps: Boundary value problems, circle domains, Fuchsian and Schottky uniformization

**Authors: **
Bobenko, A. I. and
Sechelmann, S. and
Springborn, B.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Note: **Preprint

**Date: **
2016

**Download: **
internal

###### Numerical Methods for the Discrete Map $Z^a$

**Authors: **
Bornemann, F. and
Its, A. and
Olver, S. and
Wechslberger, G.

**In Collection: **
Advances in Discrete Differential Geometry, Springer

**Date: **
2016

**Download: **
arXiv

###### Discrete Riemann surfaces based on quadrilateral cellular decompositions

**Authors: **
Bobenko, A. I. and
Günther, F.

**Date: **
2015

###### Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns

**Authors: **
Bobenko, A. I. and
Dimitrov, N. and
Sechelmann, S.

**Date: **
2015

**Download: **
arXiv

###### Hyper-ideal Circle Patterns with Cone Singularities

**Author: **
Dimitrov, N.

**Journal: **Results in Mathematics,
pages 1-45

**Date: **
2015

**DOI: **
10.1007/s00025-015-0453-3

**Download: **
external
arXiv

###### The asymptotic behaviour of the discrete holomorphic map $Z^a$ via the Riemann-Hilbert method

**Authors: **
Bobenko, A. I. and
Its, A.

**Journal: **Duke Math.~J.

**Note: **accepted

**Date: **
2015

**Download: **
arXiv

###### Discrete Riemann surfaces: linear discretization and its convergence

**Authors: **
Bobenko, Alexander and
Skopenkov, Mikhail

**Journal: **J. reine und angew. Math.

**Date: **
Oct 2014

**DOI: **
10.1515/crelle-2014-0065

**Download: **
external
arXiv

###### Diskretisierung in Geometrie und Dynamik - Elastische Stäbe und Rauchringe

**Authors: **
Bobenko, A.I. and
Springborn, B.

**Journal: **Mitteilungen der DMV,
21(1):218-224

**Date: **
Dec 2013

**Download: **
external

###### There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

**Authors: **
Izmestiev, Ivan and
Kusner, Robert B. and
Rote, Günter and
Springborn, Boris and
Sullivan, John M.

**Journal: **Geometriae Dedicata,
166(1):15-29

**Date: **
Oct 2013

**DOI: **
10.1007/s10711-012-9782-5

**Download: **
external
arXiv

###### Discrete complex analysis – the medial graph approach

**Authors: **
Bobenko, Alexander I. and
Günther, Felix

**Journal: **Actes des rencontres du CIRM 3 no. 1: Courbure discrète: théorie et applications,
pages 159-169

**Date: **
2013

**DOI: **
10.5802/acirm.65

**Download: **
external

#### Team+

#### Prof. Dr. Alexander I. Bobenko +

**Projects: **
A01,
A02,
C01,
B02,
CaP

**University: **
TU Berlin,
Institut für Mathematik,
MA 881

**Address: **Strasse des 17. Juni 136, 10623 Berlin, Germany

**Tel: **+49 (30) 314 24655

**E-Mail: **
bobenko[at]math.tu-berlin.de

**Website: **http://page.math.tu-berlin.de/~bobenko/

#### Dr. Ulrike Bücking +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
buecking[at]math.tu-berlin.de

**Website: **http://page.math.tu-berlin.de/~scheerer/

#### Prof. Dr. Boris Springborn +

**Projects: **
A01,
A11

**University: **
TU Berlin

**E-Mail: **
springb[at]math.TU-Berlin.DE

**Website: **http://page.math.tu-berlin.de/~springb/

#### Niklas Affolter +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
affolter[at]math.tu-berlin.de

#### Dr. Felix Günther +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
fguenth[at]math.tu-berlin.de

#### Hana Kourimska +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
kourim[at]math.tu-berlin.de

#### Isabella Retter +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
thiesen[at]math.tu-berlin.de

#### Dr. Stefan Sechelmann +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
sechel[at]math.tu-berlin.de

#### Lara Skuppin +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
skuppin[at]math.tu-berlin.de

#### Dr. Ananth Sridhar +

**Projects: **
A01

**University: **
TU Berlin

**E-Mail: **
sridhar[at]math.tu-berlin.de