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# Isoparametric surfaces in $\mathbb{E}(\kappa,\tau)$-spaces

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.

Seminario 1ª planta, IEMath-GR

# Constant mean curvature surfaces in $\mathbb{E}(\kappa,\tau)$-spaces

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.

# 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.

# Compact stable surfaces with constant mean curvature in Killing submersions

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.

Seminario 1ª Planta, IEMath-GR

# On the area growth of constant mean curvature graphs in $\mathbb{E}(\kappa,\tau)$-spaces.

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.

Seminario 1ª planta, IEMath

# Superficies totalmente umbilicales en grupos de Lie métricos tridimensionales

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.

Seminario de Matemáticas. 1ª Planta. Sección de Matemáticas

# Nuevos ejemplos de superficies de curvatura media constante en $M^2\times \mathbb{R}$ (II)

Seminario de Matemáticas. 1ª Planta, sección de Matemáticas.

# Finales anulares para funciones armónicas

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.

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

# Número de grafos minimales

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.

Seminario de Matemáticas. 2ª Planta, sección de Matemáticas.

# José M. Manzano

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