# Participants and talks

Browse the talks and speakers who participated in our summer 2018 workshop at Technische Universität München (TUM).

## Alexander Bobenko

## Karsten Große-Brauckmann

**From triply periodic minimal surfaces to networks**

Triply periodic surfaces are familiar from modelling real world interfaces. These can be polymer systems or butterfly wings which produce structural colours. The functional to be minimized can be area, Willmore, or a Helfrich functional. The most common such triply periodic surface is the so-called gyroid. Nevertheless, as yet there is no satisfying answer as to what makes this surface optimal. In my talk, I state a problem which distinguishes the gyroid from \emph{all} other triply periodic surfaces. It is a length minimization problem for periodic graphs; the graphs are familiar from various constructions of minimal and constant mean curvature surfaces. As a theorem, the unique minimizer is the gyroid graph. (Joint with Jerome Alex).

## Ulrike Bücking

**Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices**

Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete conformal map. We study discrete conformal maps defined on parts of a triangular lattice $T$ with strictly acute angles.

That is, $T$ is an infinite triangulation of the plane with congruent strictly acute triangles. We show that a smooth conformal map $f$ can be approximated on a compact subset by such discrete conformal maps $f^\varepsilon$, defined on a part of $\varepsilon T$ for $\varepsilon>0$ small enough, and this convergence is in fact in $C^\infty$. Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of $f$. Finally, we relate these discrete conformal maps to discrete minimal surfaces using a discrete Weierstrass representation by Wai Yeung Lam.

## Fran Burstall

**Discrete omega surfaces**

Omega surfaces, discovered by Demoulin in 1911, comprise a rich class of surfaces of classical interest, including linear Weingarten surfaces, isothermic and Guichard surfaces. They are an integrable system with a duality, Darboux transforms and a spectral deformation all of which can be traced back to their relation with isothermic surfaces in the Lie quadric.

In this talk I shall sketch a satisfying discretization of this theory which retains all the details of the classical story. Along the way, we will present a novel reformulation (and mild generalization) of the Bobenko-Pinkall theory of isothermic nets.

## Hao Chen

**New TPMS of genus 3, and where to find them**

I will present some new triply periodic minimal surfaces of genus three (TPMSg3). More specifically, we prove the existence of two “exotic” 2-parameter families of TPMSg3. They can be seen as orthorhombic deformation families, one from Schwarz’ D surface, the other from Schwarz’ P or H surface. But they are different from any of the previous deformations. In particular, they are not part of Meeks’ 5-parameter family of TPMSg3, although they do join Meeks’ family on their boundaries. This is a joint work with Matthias Weber.

## Josef Dorfmeister

**Minimal Legendrian immersions in S^5, harmonic Gauss maps and loop groups**

We will start from arbitrary immersions from a contractable immersion into CP^2 without complex points and will discuss the moving frame equations. This will lead to a local characterization of minimal surfaces and minimal Lagrangian surfaces by primitive harmonic maps. This implies a loop group method for the construction of all such surfaces. Finally we will discuss how one can extend the theory presented above to a construction method of all minimal Legendrian surfaces into S^5 and what this means for the construction of all minimal Lagrangian surfaces in CP^2.

## María Ángeles García-Ferrero

**Minimal graphs with prescribed level set**

Combining suitable local solutions and global approximation theorems we can construct solutions to PDEs with prescribed properties. I will apply this strategy with some modification to show the existence of minimal graphs on the n-dimensional unit ball whose transverse intersection with any horizontal hyperplane (equivalently, level set) is diffeomorphic to any hypersurface satisfying some mild conditions. This implies that there is not a uniform bound for the (n − 1)-measure of the zero level set of nontrivial minimal graphs.

This is a joint work with A. Enciso and D. Peralta-Salas.

## Udo Hertrich-Jeromin

**Minimal surfaces and Quadrics**

We shall discuss an obscure relation between certain minimal surfaces and quadrics. In the process, various interesting and novel facts or observations about minimal or maximal surfaces in a Euclidean or Minkowski ambient geometry, and about their respective Weierstrass representations, are revealed.

## Shimpei Kobayashi

**Minimal surfaces in the 3-dimensional Heisenberg group**

Constant mean curvature surfaces in 3-dimensional homogeneous spaces have been intensively studied during the last decades. In this talk I will give a loop group formulation for minimal surfaces in the 3-dimensional Heisenberg group. This is a joint work with Josef Dorfmeister and Jun-ichi Inoguchi.

## Wayne Rossmann

**Singularities on discrete omega surfaces**

The class of omega surfaces maintains certain mathematical structures of the subclass of isothermic surfaces, but is a much broader collection. While isothermic surfaces in 3-dimensional spaceforms generally lack singularities, omega surfaces that project to non-isothermic surfaces in spaceforms, including a number of surface classes recently receiving attention, typically do produce singularities. Upon discretizing omega surfaces, one then expects to see discrete analogs of singularities. Without continuity, discrete omega surfaces are in some sense singular everywhere, so we need to first specify what we mean by "singularity" in this case, and we then follow up with related results.

## Gudrun Szewieczek

**Discrete channel surfaces**

Channel surfaces, envelopes of a 1-parameter family of spheres, were studied by classical geometers using different geometric data. They can be characterized by their curvature sphere congruences or Lie cyclides and come along with particular spheres, namely Blaschke’s quer-spheres. In the talk we give a short introduction to results about smooth channel surfaces from the viewpoint of Lie sphere geometry, which are then used to define discrete channel surfaces. In this way, we obtain particular discrete Legendre maps with various geometrie data living on edges, vertices and faces of the domain of the Legendre map. We discuss the interplay of these geometric objects and, conversely, investigate which prescribed data allows the reconstruction of a (unique) discrete channel surface. In this realm, we also characterize discrete isothermic channel surfaces and reveal the subclass of discrete Dupin cyclides.

## Martin Traizet

**Gluing constructions for constant mean curvature surfaces**

CMC-1 surfaces in euclidean space admit a Weierstrass-type Representation discovered by Dorfmeister, Pedit and Wu and called the DPW method. I will explain how to use this method to construct examples which can heuristically be described as gluing together spheres and pieces of Delaunay surfaces. Such surfaces have previously been constructed by Kapouleas and Mazzeo-Pacard using PDE methods. I will explain a strategy to choose the DPW potential. The Monodromy Problem is solved by an Implicit Function argument.

## Masashi Yasumoto

**Discrete timelike minimal surfaces**

In the smooth case, timelike minimal surfaces are highly related to linear and nonlinear wave equations. In fact, each coordinate function of a timelike minimal surface solves a 1D wave equation, and the metric function of a timelike minimal surface gives a solution to a nonlinear wave equation.

In this talk we will describe a theory for discrete timelike minimal surfaces in Minkowski 3-space. First we introduce a Weierstrass-type representation for them, and we show that each coordinate function of a discrete timelike minimal surface satisfies a discrete version of the 1D wave equation. This result can be regarded as a ”reparametrization” of discrete timelike minimal surfaces, providing another representation formula. Finally, we will discuss the possibility of convergence of discrete timelike minimal surfaces to the smooth counterparts, and we note that there are examples of discrete timelike minimal surfaces that do not converge to the smooth counterparts.