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# Lawson correspondence for minimal and constant mean curvature surfaces

## Pascal Romon Université Paris-Est Marne-la-Vallée

Lawson correspondence between minimal surfaces in $\mathbb{S}^3$ and constant mean curvature (CMC) surfaces in $\mathbb{R}^3$ is instrumental in building examples of surfaces, but also reveals a lot about the structure of underlying PDEs. In discrete geometry, simple definitions of CMC surfaces are not obvious (many have been given) and an analog to Lawson correspondence has eluded researchers for a long time. We will present here a natural and computation-friendly definition, based on Lax pairs, which showcases the analogy between the two types of surfaces, as well as proposes a construction method. We will try to outline the similarities and differences between the smooth and discrete realms. Joint work with Alexander Bobenko (TU Berlin)

Seminario primera planta, IEMATH

# Constant mean curvature surfaces in discrete geometry

## Pascal Romon Université Paris-Est Marne-la-Vallée

Defining a relevant notion of constant mean curvature and constant mean curvature surfaces is difficult in discrete geometry. In order to make sense of these definitions, we will first recall what properties are shared by the smooth CMC surfaces, e.g. criticality, associated family, integrable system PDE, that we would like to hold also in the discrete case. Then we will describe how such notions can or cannot be carried out in the discrete case. (Joint work with Sasha Bobenko)

Seminario 1ª planta, IEMath

# Discrete Differentil Geometry, I

## Pascal Romon Université Paris-Est Marne-la-Vallée

Topics: Introduction and basic concepts Discrete curves in the plane. Aproximation properties Discrete surfaces. Mean curvature flow. Discrete Gauss-Bonnet theorem

Seminario 2ª Planta, IEMath-GR

# Killing spinors and Weierstrass representations of surfaces in 3 and 4 dimensions

## Pascal Romon Université Paris-Est Marne-la-Vallée

In the last two decades a variety of formulae have been proposed to parametrize conformal immersions of surfaces in 3 or 4 dimensions, often named "Weierstrass formula", referring to the classical formula of Weierstrass and Enneper for minimal surfaces. Most of these expressions rely on "spinors" satisfying a "Dirac equation" (Taimanov, Konopelchenko, Kusner, Schmitt etc.). We will explain where these formulae come from and how they are related to more recent work on classical spinors by Bär, Friedrich and Roth. If time allows, I will also mention the point of view of isometric immersions and moving frames.

Seminario 1ª Planta

# Pascal Romon

## Université Paris-Est Marne-la-Vallée

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Francia