# m:iv seminar archive

## K Leschke & F Neumann: jReality, 3D-XplorMath and SURFER

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from 04:00 PM to 06:00 PM

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We give an introduction how the 3D projector in the VisLab can be used for 3D viewing of surfaces.

## José M. Manzano : Compact embedded minimal surfaces in S2xS1

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from 04:00 PM to 06:00 PM

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**Abstract:**

We will begin by surveying some topological obstructions for compact surfaces to be embedded minimally in different ambient three-manifolds. In the case of the Riemannian product S^2xS^1, we will show that a compact surface can be embedded minimally in S^2xS^1 if and only if it has odd Euler characteristic.To illustrate this result, we will construct compact embedded minimal surfaces in S^2xS^1 with a high number of symmetries, first by means of solutions to the Plateau problem with respect to suitable contours, and second by the so-called conjugate Plateau construction. Time permitting, we will explain how this conjugate technique can be extended to obtain constant mean curvature surfaces in H^2xR and S^2xR with prescribed symmetries. This is based on a joint work with Julia Plenhert and Francisco Torralbo.

**links**: https://arxiv.org/abs/1104.1259 and https://arxiv.org/abs/1311.2500

*We will go to lunch with Miguel at 12:30pm.*

## Fran Burstall: Introduction to discrete isothermic surfaces

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from 11:00 AM to 12:00 PM

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## Liam Collard : The Weierstrass representation for minimal surfaces

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from 11:00 AM to 12:00 PM

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## Francesca Tripoli: On the topology of surfaces with the generalised simple lift property

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from 11:00 AM to 12:00 PM

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## Simon Cox: The mathematics of soap bubbles

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from 11:00 AM to 12:00 PM

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## Dr Ross Ogilvie, Moduli of harmonic tori in S^3.

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Jul 22, 2017 12:00 PM

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## Josef F. Dorfmeister

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from 02:00 PM to 03:00 PM

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## K Grosse-Brauckmann

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from 02:00 PM to 03:00 PM

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Title:

## Keomkyo Seo (Sookmyung University, Korea) "Necessary conditions for submanifolds to be connected in a Riemannian manifold"

Abstract: It is well-known that any simple closed curve in ℝ3 bounds at least one minimal disk, which was independently proved by Douglas and Radó. However, for any given two disjoint simple closed curves, we cannot guarantee existence of a compact connected minimal surface spanning such boundary curves in general. From this point of view, it is interesting to give a quantitative description for necessary conditions on the boundary of compact connected minimal surfaces. We derive density estimates for submanifolds with variable mean curvature in a Riemannian manifold with sectional curvature bounded above by a constant. This leads to distance estimates for the boundaries of compact connected submanifolds. As applications, we give several necessary conditions and nonexistence results for compact connected minimal submanifolds, Bryant surfaces, and surfaces with small L2 norm of the mean curvature vector in a Riemannian manifold.

## Prof. Fran Burstall (University of Bath): Linear Weingarten surfaces in Lie sphere geometry

## Prof. Joeri Van Der Veken

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from 02:00 PM to 03:00 PM

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Title: Lagrangian submanifolds of S3 x S3

## m:iv Mini-Workshop 2018 Leicester

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from 10:00 AM to 05:00 PM

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**Speakers**:

**Joseph Cho (Kobe)**

**Title: Discrete omega surfaces - a rough introduction**

**Abstract:**Omega-surfaces were discovered by Demoulin, and one of the motivations of studying Omega-surfaces arise from the fact that well-known classes of surfaces such as constant mean curvature surfaces, constant Gaussian curvature surfaces, and linear Weingarten surfaces are all examples of Omega-surfaces. Demoulin showed that Omega-surfaces are the class of surfaces that envelop a pair of isothermic sphere congruences, allowing one to apply the rich theory of isothermic surfaces to examine the properties of Omega-surfaces. Recent interest in integrable systems has led to renewed interest in Omega-surfaces, exemplified in works by Burstall, Cecil, Ferapontov, Hertrich-Jeromin, Musso, Nicolodi, etc..

**Joshua Cork (Leeds)**

** ****Title**: Symmetric calorons and the rotation map.

**Abstract**: For quite some time, there has been much interest in Yang-Mills instantons. These are manifested as finite-action anti-self-dual connections over some four manifold. They possess a very rich geometry, providing examples of hyperkahler metrics, and various applications to physics. A great deal of the progress in understanding the most basic examples of these objects has been through studying fixed points under isometries. In this talk, I will discuss calorons, which are instantons on S^1\times R^3. In particular, I shall present a classification result of cyclically symmetric calorons, where the cyclic groups considered are coupled to a non-spatial isometry known as the rotation map.

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Title: DISPERSIONLESS INTEGRABLE SYSTEMS IN 3D AND EINSTEIN-WEYL GEOMETRY
(based on joint work with Boris Kruglikov)
**

**Jenya Ferapontov (Loughborough)**

**Abstract**

For several classes of second-order dispersionless PDEs, we show that their characteristic varieties define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the method of hydrodynamic reductions. This demonstrates that the integrability of dispersionless PDEs can be seen from the geometry of their characteristic varieties.

**Gerasim Kokarev (Leeds)**

**Title:**Conformal volume, eigenvalue problems, and related topics

**I will give a short survey about the classical inequalities for the first Laplace eigenvalue on Riemannian manifolds, such as the Reilly inequality, the Li-Yau inequality, and tell about related history and questions. I will then discuss results concerning their versions for the higher Laplace eigenvalues and concerning estimates for the number of negative eigenvalues of Schrodinger operators.**

**Abstract:**

**
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**Yuta Ogata (Okinawa)**

**Title:** Constant mean curvature surfaces and positon-like solutions

Abstract:The classical Bianchi-Baecklund transformation for constant mean curvature surfaces in Euclidean 3-space has been studied by many researchers. In this talk, we introduce the method to construct positon-like solution of elliptic sinh-Gordon equation via successive Bianchi-Baecklund transformations with a single spectral parameter. We also show the recipe of the corresponding constant mean curvature surfaces of positon-like solutions.

Gudrun Szewieczek (Wien)

**Title**: Combescure transformations – a helpful tool for the study of Guichard nets

**Abstract** : Two surfaces are Combescure transformations of each other if they have parallel tangent planes along corresponding curvature lines. The existence of particular Combescure transformations can be used to characterize various classes of integrable surfaces, e.g. the Christoﬀel transformation for isothermic surfaces. In this talk we investigate this concept for triply orthogonal systems and discuss the subclass of Guichard nets, which arise as special coordinate systems of 3-dimensional conformally ﬂat hypersurfaces.

## Wei Yeung Lam

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from 02:00 PM to 03:00 PM

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## M Padilla

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from 02:00 PM to 03:00 PM

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## B Nelli

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from 02:00 PM to 03:00 PM

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**Title:** Hypersurfaces with constant higher mean curvature

**Abstract:** We give an overview of some old and new results

about the shape of hypersurfaces whose one symmetric function

of the principal curvatures is constant.

## L Ferrer

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from 02:00 PM to 03:00 PM

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**Title**: Minimal surfaces in ℍ

^{2}× ℝ

**Abstract**: One of the most interesting advances of the field of minimal surfaces in the

^{2}× ℝ, obtained jointly with F. Martin, R. Mazzeo and M. Rodriguez. In order to

^{2}× ℝ. We recall definitions,

## T Wani

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from 02:00 PM to 03:00 PM

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Abstract: In this talk, I will discuss the main tools of quaternionic calculus on a Riemann surface and use it to connect the theory of conformal minimal immersions from a Riemann surface into the Euclidean 3-space to the theory of integrable systems. This connection will helps us to use the tools and techniques of Integrable Systems in studying the minimal immersions. At the end, I will discuss the technique of dressing operation to generate new harmonic maps and new minimal immersions.

## Graham Smith

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from 03:00 PM to 05:00 PM

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Title: On the asymptotic geometry of finite-type $k$-surfaces in $3$-dimensional hyperbolic space.Abstract: We define a finite-type $k$-surface to be a complete, immersed surface of finite area and of constant extrinsic curvature equal to $k$. In earlier work, we showed that, for all $k\in]0,1[$, every finite-type $k$ surface has finite many, $N$ say, ends, and that that the space of finite-type $k$-surfaces in $3$-dimensional hyperbolic space is homeomorphic to the space of pointed ramified covers of the Riemann sphere. In this talk we study the asymptotic structure of the ends of such surfaces, showing that each such end is asymptotic to a unique preferred geodesic, which we call the Steiner geodesic. To each finite-type $k$-surface is thus associated $N$ Steiner geodesics, and we show that the vector consisting of all end-points of these geodesics defines a lagrangian immersion from each stratum of the space of finite-type $k$-surfaces into a suitable open subset of the Cartesian product of $2N$ copies of the Riemann sphere.