In this talk, we will consider an arbitrary orientable Riemannian surface $M$ and an open relatively compact domain $\Omega\subset M$ with piecewise regular boundary. Given a Killing submersion $\pi:\mathbb{E}\to M$, we will discuss some properties of the divergence lines spanned by a sequence of minimal graphs over $\Omega$, as well as how they produce certain laminations in $\pi^{-1}(\Omega)$ whose leaves are vertical surfaces (after considering a subsequence). We will apply these results to give a general solution to the Jenkins-Serrin problem over $\Omega$ under natural necessary assumptions. This is a joint work with Andrea del Prete and Barbara Nelli.

Watch the talk in youtube

In this talk, we will describe the 1-parameter family of horizontal Delaunay surfaces in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ with supercritical constant mean curvature. These surfaces are not equivariant but singly periodic, and they lie at bounded distance from a horizontal geodesic. We will show that horizontal unduloids are properly embedded surfaces in $\mathbb H^2\times\mathbb{R}$. We also describe the first non-trivial examples of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. They have constant mean curvature $H>\frac{1}{2}$. Finally, we prove that there are no properly immersed surface with critical or subcritical constant mean curvature at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times\mathbb{R}$.

Acceso a sala Zoom.

We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group:

• isoparametric surfaces,
• surfaces with constant principal curvatures,
• extrinsically homogeneous surfaces,
• $H$-surfaces with vanishing Abresch-Rosenberg differential.

This is a joint work with Miguel Domínguez Vázquez.

This is the second and last part of a 6-hour PhD mini-course on constant mean curvature surfaces in $\mathbb{E}(\kappa,\tau)$-spaces.

In this 6-hour PhD minicourse, we will give an introduction to constant mean curvature surfaces in simply-connected Riemannian homogeneous three-manifolds with four-dimensional isometry group. These spaces are contained in a two-parameter family $\mathbb{E}(\kappa,\tau)$, depending on two real parameters $\kappa,\tau\in\mathbb{R}$, which includes the space forms $\mathbb{R}^3$ and $\mathbb{S}^3$, the product spaces $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$, and the Lie groups $\mathrm{Nil}_3$, $\mathrm{SU}(2)$ and $\mathrm{SL}_2(\mathbb{R})$ equipped with special left-invariant metrics.

Throughout the course, we will discuss some basic facts about the geometry of $\mathbb{E}(\kappa,\tau)$-spaces, as well as some of the most important results for constant mean curvature surfaces immersed in them. Among other topics, we will review the existence of harmonic maps and holomorphic quadratic differentials, the isometric correspondence and the conformal duality, the conjugate Plateau constructions, the Jenkins-Serrin problem, and the solution to the Bernstein problem for surfaces with critical mean curvature.

A Killing submersion is a Riemannian submersion from an orientable 3-manifold to an orientable surface, such that the fibres of the submersion are the integral curves of a Killing vector field without zeroes. The interest of this family of structures is the fact that it represents a common framework for a vast family of 3-manifolds, including the simply-connected homogeneous ones and the warped products with 1-dimensional fibres, among others. In the first part of this talk we will discuss existence and uniqueness of Killing submersions in terms of some geometric functions defined on the base surface, namely the Killing length and the bundle curvature. We will show how these two functions, together with the metric in the base, encode the geometry and topology of the total space of the submersion. In the second part, we will prove that if the base is compact and the submersion admits a global section, then it also admits a global minimal section. This gives a complete solution to the Bernstein problem (i.e., the classification of entire graphs with constant mean curvature) when the base surface is assumed compact. Finally we will talk about some results on compact orientable stable surfaces with constant mean curvature immersed in the total space of a Killing submersion. In particular, if they exist, then either (a) the base is compact and it is one of the above global minimal sections, or (b) the fibres are compact and the surface is a constant mean curvature torus.

In this talk we will discuss some estimates for the extrinsic area growth of constant mean curvature graphs in the simply-connected homogenous 3-manifolds $\mathbb{E}(\kappa,\tau)$, whose isometry group has dimension at least 4. Such estimates follow from analyzing the height that geodesic balls reach in $\mathbb{E}(\kappa,\tau)$, and will allow us to give sharp upper bounds for the extrinsic area growth of distinguished families of constant mean curvature surfaces such as invariant surfaces, complete graphs and $k$-noids. Finally we will focus on the study of entire minimal graphs in $\mathbb{E}(\kappa,\tau)$ with $\kappa<0$, for which sharper estimates are obtained by assuming restrictions on the height growth. This is a joint work with Barbara Nelli, which can be downloaded at http://arxiv.org/abs/1504.05239.

En esta charla, analizaremos y clasificaremos las superficies totalmente umbilicales en los grupos de Lie métricos tridimensionales, tanto en el caso unimodular como en el no-unimodular.
Este es un trabajo conjunto con Rabah Souam.

Un final anular E de una superficie de Riemann M se dice de tipo finito para una función armónica $f:E\rightarrow \mathbb{R}$ si existe un conjunto de nivel de $f$ que tiene un número finito de finales como complejo 1-dimensional. Veremos que en estas condiciones, podemos decidir a partir de los conjuntos nodales de $f$ si la estructura conforme de E es parabólica o hiperbólica. En el caso de que M sea una superficie mínima propiamente inmersa en $\mathbb{R}^3$ y $f$ una función coordenada descartaremos el caso hiperbólico, deduciendo que si existe un plano de $\mathbb{R}^3$ que corta a un final anular de M en un complejo 1-dimensional con un número finito de finales, entonces el tipo conforme de ese final es parabólico (este resultado se usará en la charla del día 28) y que si en vez de uno tenemos dos planos no paralelos con esa propiedad, entonces el final es de curvatura total finita.

El objetivo de esta charla es estudiar el número de componentes que puede tener un dominio en una variedad riemanniana sobre el que puede definirse un grafo minimal que se anule en la frontera. En esta línea, comentaremos los resultados de V. Tkachev y H. Rosenberg que acotan dicho número de componentes a un máximo de tres en el caso del plano $\mathbb{R}^2$ y en otros casos de curvatura de Ricci no negativa.