# Participants and talks

Browse the talks and speakers who participated in our Autumn 2019 workshop at the University of Leicester.

## Fran Burstall

**Curved flats and O-systems**

We show that the O-surfaces of Konopelchenko-Schief and their associated Combescure transforms are a special kind of curved flat in the sense of Ferus-Pedit. This gives two view points on a number of well-known interesting integrable systems such as isothermic surfaces, omega surfaces, 3-dimensional Guichard nets along with some less studied examples such as G-surfaces (Calapso, 1920) and a non-classical geometry of null line-congruences in a 5-dimensional quadric. This is a report on work in progress with Pember and Siewiezcek among others.

## Emma Carberry

**Deformations of Spectral Curves and Applications to Constant Mean Curvature and Harmonic Tori**

I shall describe a number of geometric results which follow from considering deformations of spectral curve data. For example, constant mean curvature (CMC) tori in the 3-sphere are dense amongst CMC planes of finite type. By contrast, the closure of the space of CMC tori in Euclidean 3-space forms a rather interesting set, the points of highest dimension in an irreducible real algebraic variety which contains smooth points of different dimensions. The Euclidean case is substantially more difficult than the spherical one, and includes the development of a local deformation theory for a curve together with one or two meromorphic differentials, whose periods should be preserved under deformation. In particular, such data possesses a universal local deformation. This theory was developed in a very general situation in order that it can be applied in other contexts. I shall also discuss the space of (spectral curves of) harmonic tori in the 3-sphere. This is generically 2-dimensional and for spectral genus one its path-connected components have the structure of a ”helecoid" or an annulus.

## Joseph Cho

**Discretization principle via polynomial conserved quantities**

In this talk, we look at polynomial conserved quantities of flat connections coming from the isothermicity of a surface. In particular, we review how they behave under Darboux transformations and relate them to the discrete isothermic surface theory.

## Sebastian Heller

**Area Estimates for High genus Lawson surfaces via DPW**

The DPW method constructs minimal surfaces in the 3-sphere in terms of families of meromorphic connections, satisfying a reality condition (intrinsic closing) and an extrinsic closing condition. Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces _m;k of high genus by deforming the corresponding DPW potential. As a by product of our method we show the area estimate:

Area(_m;k) = 4_(m + 1)_1 ï¿½_m2(k + 1)+ O_1(k + 1)3__for m _xed and k large, where _m = (m+1)R 0 (1ï¿½xm)21ï¿½x2m+2 dx:

The talk is based on joint work with L. Heller and M. Traizet

## Tim Hoffmann

**On discrete geodesics**

This talk will be an overview on discrete notions of geodescis and geodesic nets in discrete differential geometry. There have been notions of geodesics on triangulated surfaces for quite a while but geodesics as parameterlines in special parametrizations have been rarely considered in discrete differenatial geometry. However, they do show up in discrete Voss surfaces and in the focal surfaces of curvature line parametrized nets and the fact that geodesics are curves that are invariant under isometries helps to define a discrete notion of developable surfaces.

## Martin Kilian

**Properly embedded minimal annuli in S2 x R**

This talk is about the classification of properly embedded minimal annuli in S2 x R, and confirms a conjecture by Harold Rosenberg that all such surfaces are foliated by circles. This is joint work with Laurent Hauswirth and Martin Schmidt.

## Katrin Leschke

**Links between the integrable systems of a CMC surface**

A CMC surface in 3-space is constrained Willmore and isothermic. It is well known that these 3 surface classes are each determined by a family of flat connections. In this talk we discuss links between the corresponding families of flat connections: we show that parallel sections of the associated family of flat connections of the harmonic Gauss map give algebraically the parallel sections of the other families. In particular, we obtain links between transformations of CMC surfaces, isothermic surfaces and constrained Willmore surfaces which are given by parallel sections, such as the associated family, the simple factor dressing and the Darboux transformation.

## Francisco Martin

**New examples of translating solitons of the Mean Curvature Flow**

The purpose of this talk is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in \mathbb{R}3, particularly those which are graphs over domains in the Euclidean plane.

## Ian McIntosh

**Equivariant minimal surfaces in real and complex hyperbolic spaces**

I will summarise the work John Loftin and I have done over the last few years regarding the moduli spaces of equivariant minimal surfaces in real and complex hyperbolic spaces. For the most part this exploits so-called nonabelian Hodge correspondence between equivariant (or "twisted") harmonic maps into non-compact symmetric spaces and Higgs bundles. There are still a lot of interesting questions regarding properties of these minimal surfaces and part of my aim will be to get to the point where I can describe some of the questions I would like to answer about these.

## Yuta Ogata

**Ribaucour transforms and their singularities**

The Ribaucour transformation is a generalization of Darboux transformations and has been studied by many researchers. In this talk, we introduce our result related to Ribaucour transformation for surfaces of revolution. We also study the singularities of them.

## Franz Pedit

**Commuting Hamiltonian Flows on Space Curves**

In this lecture we will discuss the structure of the infinite dimensional manifold of space curves as a phase space of an infinite dimensional completely integrable system. This manifold admits an infinite hierarchy of energy functionals, for instance, length, total torsion, elastic energy, projected area etc., whose Hamiltonian flows are avatars of the non-linear Schrödinger hierarchy. Consistently working within the geometric setting of space curves, rather than with their derived invariants such as curvature, will clarify the role of the various ingredients of infinite dimensional integrable systems: loop algebras and groups, zero curvature equations, Lax pairs, spectral curves, stationary solutions, and Darboux transformations. In addition to new characterizations of elastic curves via isoperimetric conditions, our geometric setup can conceivably be extended to surfaces in 4-space.

## Wayne Rossman

**Discrete surfaces with topology and/or singularities**

Global behaviour of smooth surfaces, such as topological type and existence of singularities, has been well studied in cases with particular curvature properties. The goal of this talk is to discuss what similar global behaviours can be explored on discrete surfaces, and is based on joint works with Masashi Yasumoto (and material from his talk will be referred to), Joseph Cho, Seong-Deog Yang, Tim Hoffmann, Takeshi Sasaki and Masaaki Yoshida.

## Thomas Raujouan

**Constant mean curvature n-noids in hyperbolic space**

The DPW method gives a Weierstrass-type representation of constant mean curvature surfaces in threedimensional space forms. Using the DPW method in hyperbolic space, we will show how to construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends.

## Eduardo Mota Sanchez

**Constant Mean Curvature Surfaces and Heun's Differential Equations**

The generalised Weierstrass representation for surfaces with constant mean curvature allows to describe any conformal constant mean curvature immersion in R3, H3 or S3 with four ingredients: a Riemann surface, a base point, a holomorphic loop Lie algebra valued 1-form and the initial condition for a 2x2 linear system of ODEs. Associating to this linear system a second order differential equation from the class of Heun's Differential Equations, we can prescribe certain kind of singularities in the constructing method that appear in the resulting surface. Regular singularities produce asymptotically Delaunay ends in the surface and irregular singularities produce irregular ends. We discuss global issues such as period problems and asymptotic behavior involved in the construction of this kind of surfaces. Finally we show how to construct new parametric families of constant mean curvature surfaces in R3 with genus zero that possess at least one irregular end using these methods.

## Benjamin Sharp

**Global estimates for harmonic maps from surfaces**

A celebrated theorem of F. Hélein guarantees that a weakly harmonic map from a two-dimensional domain is always smooth. The proof is of a local nature and assumes that the Dirichlet energy is sufficiently small; under this condition it is possible to re-write the harmonic map equation using a suitably chosen frame which uncovers non-linearities with more favourable regularity properties (so-called div-curl or Wente structures). We will prove a global estimate for harmonic maps without assuming a small energy bound, utilising a powerful theory introduced by T. Rivière. This is a joint work with Tobias Lamm.

## Graham Smith

**On the elliptic sinh-Gordon equation with Durham boundary conditions**

We study the geometric significance of Durham boundary conditions for solutions of the elliptic sinhGordon equation over the compact cylinder, and we show that such solutions are always of finite type. This is joint work with Martin Killian.

## Masashi Yasumoto

**Discrete Weierstrass-type representations**

A discrete surface theory based on integrable systems has been rapidly developing by exploring higher symmetries of transformations, called multidimensional consistency. In both smooth and discrete cases, this consistency is derived from the same discrete geometric properties, and creating a discrete theory in counterpart with the smooth theory is interesting when aiming toward a unified viewpoint. In this talk we consider discrete surfaces with Weierstrass-type representations. In the smooth case, these representations for surfaces are powerful tools for constructing surfaces and analyzing their global behaviors. By the same reason, Weierstrass-type representations for discrete surfaces are important both for investigating the theory itself and for expanding our knowledge of global behaviors. We introduce how to derive the formulae in terms of transformation theory for discrete Omega surfaces, and introduce our first step in analyzing their behaviors. This talk is based on joint work with Mason Pember and Denis Polly, and further details of the latter part will be given by Wayne Rossman