# A02Discrete Parametrized Surfaces

## Developing a Theory of Discrete Surfaces with Constant Mean Curvature

In recent years, an exhaustive theory has been developed to understand and construct discrete minimal surfaces. We aim to produce something similar for the construction and classification of discrete surfaces with constant mean curvature (cmc). In particular, for discrete minimal surfaces the study of Koebe polyhedra that serve as Gauß maps has been fruitful - we are interested in analoga for discrete surfaces with constant mean curvature.

#### Scientific Details+

The aim of this project is to develop a theory of discrete surfaces of constant mean curvature (cmc surfaces for short) to an extent similar to what is known for discrete minimal surfaces. This includes understanding the structure and interconnections of various definitions of discrete cmc surfaces, their relationship with properties of their Gauß maps, construction methods, and hopefully existence and uniqueness results. Discretizing special classes of surfaces in R³ has been the starting point of many interesting developments in discrete differential geometry. Often the results have been isolated, but in some cases the the discretizations gave insight and ideas beyond its original scope and connections were made to different branches of discrete differential geometry. Discrete minimal surfaces are such a case, where among others integrable theory, combinatorics, surface discretization, and curvature theory via mesh parallelity all come together. We want to gain more insight into these interconnections by means of extending the theory - or rather complementing it - with a similar theory for cmc surfaces. In particular we plan to:

• Clarify the interrelations between several definitions of discrete cmc surfaces: There are several notions of discrete cmc surfaces all of which can be viewed as special cases of certain line congruence nets. Their interrelations are not well understood, though and the case of conical cmc surfaces has not been investigated at all.
• Investigate the structure of the Gauß maps of discrete cmc surfaces. The Gauß maps give rise to circle patterns on the sphere, which are interesting in their own right. Since the Gauß map of a smooth cmc surface is known to be harmonic, one can expect interesting analytic properties here. We hope to derive discrete versions of harmonicity for the various versions of discrete cmc surfaces.
• Find discrete analogues of the underlying partial differential equations and find analogues of Koebe's theorem for the above mentioned circle patterns: Cmc surfaces are governed by the sinh-Gordon equation and we expect to derive discrete integrable versions of the equation from the geometric definitions of discrete cmc surfaces. Likewise, we hope to find generalized versions of Koebe's theorem for the circle patterns that form the Gauß maps.
• Construction methods: It is unclear if one can expect something similar to the Weierstraß representation for (discrete) minimal surfaces. There, a variational principle allows solutions (and in the end minimal surfaces) to be constructed merely by prescribing the combinatorics. We will investigate whether one can apply similar mechanisms in case of discrete cmc surfaces. The Dorfmeister-Pedit-Wu method for cmc surfaces has been discretized for one flavour of discrete cmc surfaces, so we hope to be able to generalize that to the other variants as well.

Ideally the resulting theory should be comparable to the smooth theory and as rich and complete as the theory for discrete minimal surfaces, with (for example) its existence and uniqueness theorems.

However, the theory should not be developed just for its own sake. Not only should it be possible to extend the results to minimal surfaces in S³ via the so-called Lawson correspondence, but the intriguing interrelations between combinatorics, geometry and integrable systems - be found in the interrelations of Koebe polyhedra and discrete minimal surfaces - give hope for generalizations that not only give ways for constructing cmc surfaces but also produce insights beyond the primary agenda of this project.

One should note that this project will focus on integrable discretizations. This implies, that the we are concerned with discrete parametrized surfaces or "nets", not discretizations that come with arbitrary triangulated meshes.

#### Publications+

##### Papers
• Alexander I. Bobenko, Carl O. R. Lutz, Helmut Pottmann, and Jan Techter.
Non-Euclidean Laguerre geometry and incircular nets.
preprint, September 2020.
arXiv:2009.00978.
• Arseniy V. Akopyan, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter.
On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs.
Discrete and Computational Geometry, August 2020.
arXiv:1908.00856, doi:10.1007/s00454-020-00240-w.
• Alexander I. Bobenko and Alexander Y. Fairley.
Nets of lines with the combinatorics of the square grid and with touching inscribed conics.
preprint, November 2019.
dgd:590.
• Alexander I Bobenko, Sebastian Heller, and Nicholas Schmitt.
Minimal n-Noids in hyperbolic and anti-de Sitter 3-space.
Proceedings A of Royal Society, July 2019.
arXiv:1902.07992, doi:10.1098/rspa.2019.0173.
• Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter.
Checkerboard incircular nets: Laguerre geometry and parametrisation.
Geometriae Dedicata, April 2019.
arXiv:1808.07254, doi:10.1007/s10711-019-00449-x.
• Alexander I Bobenko, Wolfgang K Schief, Yuri B Suris, and Jan Techter.
On a Discretization of Confocal Quadrics. A Geometric Approach to General Parametrizations.
International Mathematics Research Notices, December 2018.
arXiv:1708.06800, doi:10.1093/imrn/rny279.
• Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung.
The Shape Space of Discrete Orthogonal Geodesic Nets.
ACM Trans. Graph., Vol. 37, No. 6, Article 228, November 2018.
URL: http://igl.ethz.ch/projects/dog-space/Shape-Space-Of-DOGS-SA-2018-Rabinovich.pdf, doi:10.1145/3272127.3275088.
• Zi Ye, Olga Diamanti, Chengcheng Tang, Leonidas Guibas, and Tim Hoffmann.
A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing.
Computer Graphics Forum, 37(5):93–106, August 2018.
doi:10.1111/cgf.13494.
• Michael Rabinovich, Tim Hoffmann, and Olga Sorkine-Hornung.
Discrete Geodesic Nets for Modeling Developable Surfaces.
ACM Trans. Graph., 37(2):16:1–16:17, July 2018.
doi:10.1145/3180494.
• Tim Hoffmann and Zi Ye.
A discrete extrinsic and intrinsic Dirac operator.
Preprint, February 2018.
arXiv:1802.06278.
• Arseniy Akopyan and Alexander Bobenko.
Incircular nets and confocal conics.
Transactions of the American Mathematical Society, 370(4):2825–2854, 2018.
arXiv:1602.04637, doi:10.1090/tran/7292.
• Tim Hoffmann, Andrew O. Sageman-Furnas, and Max Wardetzky.
A Discrete Parametrized Surface Theory in $\mathbb R^3$.
International Mathematics Research Notices, 2017(14):4217–4258, 2017.
arXiv:1412.7293, doi:10.1093/imrn/rnw015.
• Tim Hoffmann and Andrew O. Sageman-Furnas.
A $2 \times 2$ Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets.
Discrete & Computational Geometry, 56(2):472–501, September 2016.
arXiv:1510.06654, doi:10.1007/s00454-016-9802-6.
• Alexander I. Bobenko, Wolfgang K. Schief, Yuri B. Suris, and Jan Techter.
On a discretization of confocal quadrics. I. An integrable systems approach.
Journal of Integrable Systems, 1(1):xyw005, 2016.
arXiv:1511.01777, doi:10.1093/integr/xyw005.
• A. I. Bobenko and T. Hoffmann.
S-conical cmc surfaces. Towards a unified theory of discrete surfaces with constant mean curvature.
In A. I. Bobenko, editor, Advances in Discrete Differential Geometry. Springer, 2016.
dgd:198.
• A. I. Bobenko, T. Hoffmann, B. König, and S. Sechelmann.
S-conical minimal surfaces. Towards a unified theory of discrete minimal surfaces.
Preprint, 2015.
dgd:199.
• Alexander I. Bobenko, Udo Hertrich-Jeromin, and Inna Lukyanenko.
Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality.
Discrete and Computational Geometry, 52(4):612–629, 2014.
arXiv:1409.2001, doi:10.1007/s00454-014-9622-5.
• A. Bobenko, T. Hoffmann, and B. Springborn.
Minimal surfaces from circle patterns: Geometry from combinatorics.
Ann. of Math., 164(1):231–264, 2006.
arXiv:0305184.

##### PhD thesis
• Benno König.
A Geometric Construction for the Associated Family of S–Isothermic CMC Surfaces.
Dissertation, Technische Universität München, September 2018. Dissertation.
URL: http://mediatum.ub.tum.de?id=1368384.

#### 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/

#### Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München, Department of Mathematics, 02.06.021
Address: Boltzmannstr. 3, 85748 Garching, GERMANY
Tel: +49 89 28918384
E-Mail: tim.hoffmann[at]ma.tum.de
Website: https://geo.ma.tum.de/de/personen/tim-hoffmann.html

#### Alexander Fairley   +

Projects: A02, C01
University: TU Berlin
Tel: +49 30 31479252
E-Mail: fairley[at]math.tu-berlin.de

#### Alexander Preis   +

Projects: A02
University: TU Berlin, Institut für Mathematik, MA 886
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31429426
E-Mail: preis[at]math.tu-berlin.de

#### Jannik Steinmeier   +

Projects: A02
University: TU München, Department of Mathematics, 02.06.058
Address: Boltzmannstraße 3, 85747 Garching, GERMANY
E-Mail: jannik.steinmeier[at]web.de
Website: https://geo.ma.tum.de/de/personen/jannik-steinmeier.html

#### Dr. Jan Techter   +

Projects: A02
University: TU Berlin, Institut für Mathematik, MA 880
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424105
E-Mail: techter[at]math.tu-berlin.de
Website: https://page.math.tu-berlin.de/~techter/