We will construct approximate solutions to Riemann-Hilbert boundary value problems for null holomorphic curves in the complex Euclidean 3-space $\mathbb{C}^3$. Using this technique, we will prove that every bordered Riemann surface admits a complete proper null holomorphic embedding into a ball of $\mathbb{C}^3$, hence a complete conformal minimal immersion into $\mathbb{R}^3$ with bounded image. We will also construct properly embedded null curves in $\mathbb{C}^3$ with a bounded coordinate function; these give rise to properly embedded null curves in $SL_2(\mathbb{C})$ and to properly immersed Bryant surfaces in the hyperbolic 3-space $\mathbb{H}^3$ that are conformally equivalent to any bordered Riemann surface. In particular, we give the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type.

Recently, we have characterized together with J. Roth and P. Bayard immersions of surfaces $M$ in $\mathbb{R}^4$ by means of spinor fields, giving an spinorial analog of the Gauss–Ricci–Codazzi equations for isometric immersions. More precisely, we have shown that given a parallel spinor $\Phi$ in $\mathbb{R}^4$, its restriction to $M$ satisfies a Dirac equation $D\Phi= H\Phi$ for a Dirac operator $D$ along $M$. The difficult part lies in the converse. Given intrinsic datas: a Riemannian surface $M$, a rank 2 vector bundle $E$ on $M$, with a connection and a symmetric Evalued 2-form $B$, and additionnally a section $\Phi$ of the twisted spinor bundle $\Sigma M\otimes\Sigma E$, then $D\Phi = H \cdot\Phi$ implies (locally, i.e. on a simply connected domain) the existence of an immersion $f: M → \mathbb{R}^4$ with mean curvature $H$. In parallel, there exists a representation formula for surfaces into $\mathbb{R}^4$, known as the spinorial Weierstrass representation formula, akin to the one in $\mathbb{R}^3$, and due to Konopelchenko and Taimanov. This representation expresses any immersion as an integral over four complex valued function, satisfying a Dirac type equation. This equation was rediscovered independently by Hélein and Romon in the particular case of Lagrangian immersion. However, it remained until now somewhat unclear how these quantities were linked to spinors. In a joint work with P. Romon, we bridge the gap between these two approaches.

There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians $G_{2}(\mathbb{C}^{m+2})$ Among them, Suh classified a Hopf hypersurface $M$ in $G_{2}(\mathbb{C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of generalized Tanaka-Webster Reeb parallel Ricci tensor for $M$ in $G_{2}(\mathbb{C}^{m+2})$. Related to such a notion, we give some characterizations for a real hypersurface of Type~$(A)$ in $G_{2}(\mathbb{C}^{m+2})$.

In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ or in complex hyperbolic two-plane Grassmannians $G_2^{*}(\mathbb{C}^{m+2})$. Next by using the isometric Reeb flow we give a complete classification for hypersurfaces $M$ in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, complex hyperbolic two-plane Grassmannians $G_2^{*}(\mathbb{C}^{m+2})$ and a complex quadric $\mathbb{Q}^m$.

Revisaremos los resultados de G. Buzzard y S. Lu sobre dominabilidad del complementario en $\mathbb{C}^{2}$ de la gráfica de una función meromorfa $s:\mathbb{C}\to \mathbb{P}^{1}$. Obtendremos una familia de campos completos de tipo $\mathbb{C}^{\ast}$ en $\mathbb{C}^{2}\setminus \mathrm{graph}(s)$, y una familia de aplicaciones holomorfas de $\mathbb{C}^{2}$ a $\mathbb{C}^{2}\setminus \mathrm{graph}(s)$, sobreyectivas y con determinante de su matriz jacobiana no idénticamente cero, y definidas en términos de los flujos de estos campos completos. Veremos que la aplicación dominante dada por Buzzard y Lu es una de las aplicaciones de la familia construida. Finalmente, daremos ejemplos de cuándo el complementario en $\mathbb{C}^{2}$ de un subconjunto $A$ invariante por un campo de vectores holomorfo completo en $\mathbb{C}^{2}$ es dominable.

I will present a summary of some aspects of a study of triply-periodic minimal surfaces begun in 1966. If present plans are implemented on schedule, I will show a set of 3D-printed specimens of successively larger versions of the ‘Voronoi polyhedral gyroid’, a plane-faced approximation of the gyroid minimal surface with the same symmetry and topology. These objects are a toy model of the nucleation and growth of gyroid-like crystalline domains of the kind revealed by the Teragyroid’ computing project (cf. TeraGyroid - Final report), which employed a Lattice Boltzmann algorithm.

Cauchy's famous rigidity theorem for 3D convex polyhedra has been extended in various directions by Dehn, Weyl, A.D.Alexandrov, Gluck and Connelly. These results imply that a disk-like polyhedral surface with simplicial faces is, generically, flexible, if the boundary has at least 4 vertices. What about surfaces with rigid but not necessarily simplicial faces? A natural, albeit extreme family is given by flat-faced origamis. Around 1995, Robert Lang, a well-known origamist, proposed a method for designing a crease pattern on a flat piece of paper such that it has an isometric flat-folded realization with an underlying, predetermined metric tree structure. Important mathematical properties of this algorithm remain elusive to this day. In this talk I will show that Lang's beautiful method leads, most often, to a crease pattern that cannot be continuously deformed to the desired flat-folded shape if its faces are to be kept rigid. Most surprisingly, sometimes the initial crease pattern is simply rigid: the (real) configuration space of such a structure may be disconnected, with one of the components being an isolated point. This is joint work with my PhD student John Bowers, who also implemented a very nice program to visualize what is going on.

We obtain 2 and 3 parameter families of new minimal surfaces in the hyperbolic space $\mathbb{H}^3$, by applying Darboux transformations i.e., conformal Ribaucour transformations, to Mori’s spherical catenoids in $\mathbb{H}^3$. Depending on the values of the parameters, the minimal surfaces can have any finite number of closed curves in the boundary at infinity of $\mathbb{H}^3$ or an infinite number of such curves. In particular, we obtain minimal surfaces periodic in one variable, with certain symmetries, whose parametrization is defined in $\mathbb{R}^2 \setminus \mathcal{C}$ , where $\mathcal{C}$ is either a disjoint union of Jordan curves or a non closed regular curve. A connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of $\mathbb{H}^3$ is a closed curve. A connected unbounded domain of $\mathbb{R}^2 \setminus \mathcal{C}$ generates a non complete immersed minimal surface whose boundary at infinity consists of a finite number of closed curves. This is a joint work with Ningwei Cui.

We consider the Chern connection of a quadratic Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open $\Omega\subset M$ associated to any vector field $V$ on $\Omega$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$. Then we show some properties of the curvature tensor $R^V$ associated to $\nabla^V$ and in particular we prove that the Jacobi operator of $R^V$ along a geodesic coincides with the one given by the Chern curvature. Finally we obtain the first and the second variation of the energy functional using $\nabla^V$ and $R^V$ and we deduce some properties of Jacobi fields. The most interesting aspect o f$\nabla^V$ and $R^V$ is that allow one to make computations intrinsically as in Modern Differential Geometry, being especially interesting for Riemannian geometers that want to learn Finsler geometry.

We determine all complete projective special real surfaces. By the supergravity r-map, they give rise to complete projective special Kähler manifolds of dimension 6, which are distinguished by the image of their scalar curvature function. By the supergravity c-map, the latter manifolds define in turn complete quaternionic Kähler manifolds of dimension 16. The talk is based on joint work with Malte Dyckmanns and David Lindemann, see arXiv:1302.4570 [math.DG].

We address the problem of quantitative stability for perimeter-minimizing clusters in $\mathbb{R}^n$ with multiple volume constraints (soap bubble clusters). Our aim is to show that the perimeter deficit controls a suitable power of an asymmetry functional. A first reduction to a sequence of $\Lambda$-minimizing clusters that converge to a given minimizer is accomplished through a "selection principle" that relies on the regularity theory for clusters. Several basic questions, like the optimality of power 2 (i.e. the existence of "quadratic deformations" for any minimizing cluster), the connection between global and infinitesimal stability, and the global parametric representation of $\Lambda$-minimizers that are sufficiently close to a given minimizer, are considered. In the planar case, we prove sharp stability inequalities for standard double bubbles. Some applications and open problems will be also discussed.

Se discutirá cómo construir un ejemplo.

Relacionaremos la geometría de dominios isoperimétricos de volúmenes muy grandes con dos invariantes geométricos asociados a la teoría de superficies en un espacio homogéneo tridimensional: la constante de Cheeger y la curvatura media crítica.

There are various harmonic maps which are canonically associated to a minimal surface, e.g., the Gauss map of the immersion and the conformal Gauss map. In this talk, we will discuss how the well-known dressing operation applied to these harmonic maps is related to transformations on the minimal surface. In particular, we will show the link to the Lopez-Ros deformation, a generalised associated family and a family of Willmore surfaces given by the minimal surface. This is joint work with K. Moriya (University of Tsukuba).

A spacelike surface in a Lorentzian four-manifold is said to be marginally trapped if it mean curvature vector is null (lightlike). Marginally trapped surfaces play an important role in general relativity: they describe the horizon of black holes. However, their mathematical properties are still poorly understood, although marginally trapped surfaces satisfying several additional properties have recently been characterized: in 2006, B.-Y. Chen and F. Dillen classied marginally trapped Lagrangian surfaces in complex Lorentzian space forms; in 2007 J. Van der Veken described those marginally trapped surfaces of Lorentzian space-forms which have positive relative nullity; in 2009 the classication of marginally trapped surfaces of constant curvature in complex Lorentzian space forms was performed by B.-Y. Chen. In a joint work with Yamile Godoy (Cordoba University, Argentina), we take advantage of the natural contact structure enjoyed by the space of null geodesics of a $(n + 1)$-dimensional Lorentzian manifold, recently uncovered by B. Khesin/S. Tabachnikov and Y. Godoy/M. Salvai: it turns out that the congruence of the null geodesics normal to a $(n - 1)$-dimensional spacelike submanifold is a Legendrian submanifold. This allows to give an explicit, local description of $(n-1)$-dimensional, marginally trapped submanifolds in Lorentzian space forms, and in $\mathbb{S}^{n+1}\times\mathbb{R}$ and $\mathbb{H}^{n+1}\times\mathbb{R}$: We shall also discuss recent progress about marginally trapped surfaces in Euclidean four-space endowed with the neutral, flat metric.

Recientemente, Oihane F Blanco defendió una tesis en nuestro dpto. cuyo objetivo era clasificar los espacios lorentzianos segundo-simétricos, esto es, con $\nabla^{(2)}R := \nabla(\nabla R)=0$. En la presente charla, haré un repaso de ese resultado y del método de demostración, con especial énfasis en las técnicas elementales clásicas que estudian las implicaciones de la existencia de un campo tensorial $T$ en una variedad riemanniana con $\nabla^{(r)}T=0$ y $\nabla T \neq 0$, así como en los límites de esa aproximación.

We propose a simple proof for the vertical half-space theorem. By considering Euclidean surfaces of revolution, our approach is to construct a sub-solution for the minimal surface equation. In Heisenberg space, we find an explicit solution and obtain the theorem.

Colding y Minicozzi generalizaron la conjetura de Calabi-Yau en $\mathbb{R}^3$ y probaron que una superficie mínima, completa, embebida y con topología finita en $\mathbb{R}^3$ ha de ser propia. Coskunuzer probó que este resultado no es cierto en $\mathbb{H}^3$. En esta charla explicaré la sencilla construcción de un contraejemplo a este resultado de Colding-Minicozzi en $\mathbb{H}^2\times\mathbb{R}$, siguiendo una técnica completamente distinta a la usada por Coskunuzer en $\mathbb{H}^3$. Éste es un trabajo conjunto con Giuseppe Tinaglia.

We present results on the classification and non-existence of real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$, when the structure Jacobi operator of them satisfies conditions such as pseudo-parallellness, or $\mathcal{L}_{X}l=\nabla_{X}l$, where $X$ is orthogonal to $\xi$ or Lie recurrence. Furthermore, we present results, which complete the work that so far has been done in dimensions greater than three, when the structure Jacobi operator is $\xi$-parallel, or Lie $\mathbb{D}$- parallel or it satisfies the relation $\mathcal{L}_{\xi}l=\nabla_{\xi}l$. These results are based on a joint work with Philippos J. Xenos and Georgios Kaimakamis.

In 1991 Chen stated that every biharmonic submanifold with harmonic mean curvature vector field is minimal. From then many researchers proved the conjecture in several cases. In this talk we will see the last results concerning this conjecture.

En esta charla voy a hablar sobre un trabajo conjunto con Manuel Ritoré sobre desigualdades isoperimétricas para cuerpos convexos. No se impone ninguna hipótesis de regularidad en la frontera del cuerpo convexo. El perímetro de un conjunto perteneciente a un cuerpo convexo será el relativo a dicho cuerpo. Se van a mostrar resultados como la equivalencia entre las convergencias de Hausdorff y de Lipschitz, la convergencia del operador del perfil isoperimétrico con respecto a la convergencia de Hausdorff y la convergencia en el sentido de Hausdorff de sucesiones de regiones isoperimétricas y sus fronteras libres. También se describirá el comportamiento del perfil isoperimétrico para volúmenes pequeños y el comportamiento de regiones isoperimétricas para volumenes pequeños.

I shall first review well-known results of Simons and Schoen- Yau on stable minimal hypersurfaces in manifolds with lower curvature bounds. Then I shall describe some joint work with Vlad Moraru on an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a generalisation of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient 3-manifold are weaker than those of Heintze- Karcher but the assumptions on the surface are considerably more restrictive. I will then show how our comparison theorem provides a unified proof of various splitting theorems for 3-manifolds with lower bounds on the scalar curvature that were first proved separately by Cai-Galloway, Bray-Brendle-Neves and Nunes.

In this talk, I shall challenge the popular belief that Douglas arrived at his mysterious functional for solving the Plateau Problem by direct consideration of Dirichlet's integral and its relation to the area functional. I shall describe how, by looking at abstracts of Jesse Douglas in the Bulletin of the American Mathematical Society, I have been able to infer how Douglas MAY have arrived at his functional. Douglas was awarded one of the first Fields Medals for his work on the Plateau problem. I shall talk about some of the amusing aspects of the Fields Medal ceremony at which Douglas was awarded his prize. This is a joint work with the mathematical historian Jeremy Gray.

Se mostrará como la geometrías extrínseca e intrínseca de una superficie espacial del espacio de Lorentz-Minkowski de dimensión 4, inmersa en el cono de luz, están fuertemente relacionadas. Se incluirán varios ejemplos y métodos de construcción de tales superficies. Bajo la hipótesis de compacidad, se establecerán varios resultados que caracterizan el caso en el que la superficie es totalmente umbilical.