Two of the classical theorems in the theory of holomorphic functions are the Runge approximation theorem and the Weierstrass interpolation theorem. A complex manifold X is said to be an Oka manifold if these results, and some of their natural extensions, are valid for holomorphic maps from any Stein manifold (in particular, from complex Euclidean spaces) to X. After a brief review of the development of this subject, beginning with the classical Oka-Grauert theory and continuing with the seminal work of Gromov, I will describe some of the recent developments and future challenges in this field of complex geometry.

En esta charla vamos a presentar la variación primera y segunda del area sub-Riemanniana en 3-variedades sub-Riemannianas de contacto. Estas fórmulas permitirán encontrar una condición necesaria para que una superficie estacionaria sea estable en variedades bastantes generales. Luego pasaremos a estudiar el grupo de los movimentos rígidos del plano euclídeo RT con su estructura sub-Riemanniana (inducida por el modelo de la reconstrucción de imágenes de G. Citti y A. Sarti) y vamos a clasificar las superficies completas, estacionarias y estables inmersas en RT.
Trabajo disponible en arXiv:1109.6213v2.

In this talk, we will start with an overview of Calabi-Yau Conjecture (CYC), and its generalizations. After Meeks-Rosenberg’s lamination closure theorem, which is true for negatively curved spaces, one can expect a possible generalization of CYC to negatively curved spaces. In this talk, we will construct an example in $\mathbb{H}^3$, which shows that CYC does not extend to this case.

In this talk, we will start with an overview of the asymptotic Plateau problem, and then study the relation between being least area and being properly embedded in hyperbolic 3-space. In particular, we show that if P is an embedded least area plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.

Por resultados clásicos de Remmert, Narasimhan y Bishop, toda superficie de Riemann admite una inmersión propia y holomorfa en $\mathbb{C}^2$, y un embebimiento propio y holomorfo en $\mathbb{C}^3$. En 1990 Bell y Narasimhan conjeturaron que, de hecho, toda superficie de Riemann embebe en $\mathbb{C}^2$ de forma propia y holomorfa. Esta conjetura sigue abierta en la actualidad. En esta charla discutiremos sobre la conjetura de Bell-Narasimhan y, si el tiempo lo permite, mostraremos que la topología de la superficie no tiene influencia sobre este problema.
Trabajo conjunto con Francisco J. López disponible en arXiv:1104.1893v1 [math.CV]

Conferencia dentro del programa de doctorado Interuniversitario de Matemáticas con la colaboración del GENIL

Conferencia dentro del programa de doctorado Interuniversitario de Matemáticas con la colaboración del GENIL

Presentaré dos construcciones de origen físico que producen ciertas variedades riemannianas a partir de otras variedades riemannianas. Demostramos que las variedaes construidas de esta manera son completas si se parte de variedades completas. La composición de estas dos construcciones nos permite construir variedades cuaterniónicas kählerianas completas a partir de ciertos polinomios cúbicos.
Conferencia dentro del programa de doctorado Interuniversitario de Matemáticas

A contact metric (kappa, mu)-space is a contact Riemannian manifold such that the Reeb vector field belongs to the so-called (k,mu)-distribution. This new class of Riemannian manifolds was introduced by Blair, Koufogiorgos and Papantoniou in 1995 and then it was studied by many others. In this talk we shall point out the strong interplays between the theory of contact (kappa,mu)-manifolds and other topics, such as paracontact geometry or the theory of Legendre foliations. In particolar, we will show that any (k,mu)-manifold is canonically endowed with a so-called bi-paracontact structure and by using such geometric structure we will prove that any (k,mu)-space whose Boeckx invariant is different form 1 or -1 admits a compatible Sasakian or para-Sasakian metric.

We consider curves in $\\Rset^{2n}$ endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define $2n-1$ local differential invariants that we call symplectic curvatures of the curve. > We prove that, up to a rigid symplectic motion of $\\Rset^{2n}$, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in $\\Rset^{4}$ with constant symplectic curvatures.

In previous joint work with Alexakis, we studied the renormalized area of properly embedded minimal surfaces in hyperbolic space, which turns out to be closely related to Willmore energy. In new work, we obtain some results about how control of this renormalized area of a minimal surface leads to some regularity of its asymptotic boundary.
Esta conferencia se enmarca dentro del convenio de colaboración entre las universidades de Granada y Stanford y forma parte de la acción integrada Granada-Stanford
Seminario de conferencias del doctorado interuniversitario Matemáticas

In my PhD dissertation, I studied stable capillary surfaces with planar or spherical boundary in the absence of gravity. I will introduce both problems and our advances towards them. In the case where the boundary lies in a plane we show that the only immersed stable capillary surfaces with embedded boundary are the spherical caps. When the capillary surface lies inside the euclidean unit ball with its boundary on the unit sphere, we construct a Killing vector field for the hyperbolic metric and use it to show that if the center of mass of the region bounded between the surface and the unit sphere is at the origin, the configuration cannot be stable. As a corollary of this approach we obtain a new proof of a theorem by Barbosa and Do Carmo.

En esta charla estudiaremos algunas propiedades globales de las inmersiones armónicas de superficies de Riemann en el espacio euclidiano tridimensional. Nos detendremos con especial detalle en la geometría de las inmersiones completas y armónicas con aplicación de Gauss cuasi-conforme, y en particular, en la de aquellas con curvatura total finita. Acompañaremos estos resultados con la presentación de nuevos ejemplos de este tipo de superficies con propiedades geométricas interesantes.

Construimos nuevos ejemplos de superficies de curvatura media constante no nula en los espacios producto $\mathbb{S}^2\times\mathbb{R}$ y $\mathbb{H}^2\times\mathbb{R}$. Para ello definimos superficies mínimas apropiadas en las esferas de Berger y consideramos sus superficies hermanas.

Hasta los años ochenta, era un pensamiento bastante extendido que las superficies de tipo hiperbólico jugaban un papel marginal en la teoría global de superficies mínimas. Sin embargo las construcciones de Jorge-Xavier en 1980 y Nadirashvili en 1996 han servido de inspiración para la prueba de teoremas muy generales de existencia de superficies mínimas, completas y de tipo hiperbólico. En esta charla discutiremos las últimas técnicas de construcción de tales superficies y mostraremos algunas aplicaciones.

In this talk I will discuss the Dirichlet problem for the Laplace and minimal hypersurface PDEs with prescribed asymptotic boun- day in a Hadamard manifold M in an uni ed way by considering the family of Dirichlet problems $$\left\{ \begin{array}{ll} Q_t(u):=\mbox{div}\frac{\nabla u}{\sqrt{1+|t\nabla u|^2}}=0 &\mbox{in } M, u^{C^{\infty}(M)\cap C^0(\overline{M}),\\ u_{|\partial_\infty M}=\psi,& t\in [0,1] \end{array} \right.$$ I shall explain some results which I recently obtained in collaboration with Miriam Telichevesky. In the harmonic case a partial extension and improvements of well known results of E. Hsu and Choi. In the minimal case a partial extension of a theorem of J. A. Gálvez and H. Rosenberg by allowing the sectional curvature of M degenerate to 0 at in finity.

The metric of a Riemannian manifold $M^n$ is called locally Euclidean (l.E.) if for any point $p \\\\\\\\in M^n$ there exists a neighbourhood $U_p \\\\\\\\subset M^n$ isometric to a ball in the Euclidean space $E^n$ with the standart metric.
About manifolds and surfaces with l.E. metrics one can put many questions among which we will single the following ones:

1. Let $ds^2 = g_{ij}du^idu^j$, ($1 \\\\\\\\leq i,j \\\\\\\\leq n$) (1) be a given l.E. metric. How to find that local isometry between this metric and a ball in En the existence of which is guaranteed by the definition of a l.E. metric? What can we say about the smoothness of such an isometry?
2. How to verify is a given metric (1) locally Euclidean or not?
3. What are criteria for the existence of a global isometric immersion or embedding of a given n-dimensional l.E. metric in En?
4. What can we say about the existence/non-existence of isometric immersions and embeddings of given n-dimensional l.E. metrics in N-dimensional spaces ($N > n$) with constant curvature?
5. What is known about the local and global structure of surfaces with l.E. metrics?
6. Bendings and infinitesimal bendings of surfaces with l.E. metrics.

The geometric Cauchy problem for a class of surfaces in a 3-dimensional (pseudo)-Riemannian manifold is to construct a solution which contains a given curve and with tangent bundle prescribed along the curve. This generalizes the classical Bjoerling problem for minimal surfaces, for which there is a unique solution due to the fact that the given data is enough to determine the holomorphic Weierstrass data along this curve. Other surfaces classes which have Weierstrass representations have an analogous solution. On the other hand, flat surfaces in the 3-sphere have a solution via a dAlembert representation.
In this talk we discuss how surfaces associated to integrable systems, which have infinite dimensional versions of the Weierstrass and dAlembert representations can also be solved. Examples include spacelike and timelike CMC surfaces in Lorentz 3-space and constant negative Gauss curvature surfaces in Euclidean 3-space.

Empezamos esta charla con una breve descripción de la teoría de las aplicaciones armónicas. Esta teoría se usa para estudiar los campos vectoriales, conduciéndonos a algunos problemas variacionales. Con este enfoque extenderemos la teoría anterior a los fibrados vectoriales homogéneos, (twistor spaces) para definir la armonicidad de varias estructuras geométricas, como por ejemplo, estructuras complejas, casi de contacto y f-estructuras.