In this talk we will extend the Choi-Schoen [Invent. Math. 81 (1985) 387--394] compactness result for minimal surfaces to apparent horizons in the context of General Relativity. We also discuss some applications of the aforementioned result.

The geometry of surfaces in spaces of dimension 3, especially of the form M(c)^2 x R, has enriched in last years. An interesting problem studied in very recent papers consists of characterization and classification for constant angle surfaces in different 3-dimensional spaces belonging to the Thurston list. Such a surface is an orientable surface whose unit normal makes a constant angle with a fixed direction. When the ambient is of the form M^2 x R, the favored direction is R. It is proved that for a constant angle surface in E^3, S^2 x R or in H^2 x R, the projection of d/dt onto the tangent plane of the immersed surface, denoted by T, is a principal direction with the corresponding principal curvature identically zero. The main topic of this talk is to investigate surfaces in H^2 x R for which T is a principal direction.

The aim of this talk is to present some results about the notions of convexity for a hypersurface in a Finsler manifold. The main theorem concerns the infinitesimal and local notions of convexity which are shown to be equivalent. Using a different approach, this result extends the classical Bishop\\\\\\\'s one for the Riemannian case to the Finsler setting. It also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.

Esta es la primera de dos charlas sobre superficies con CMC en grupos de Lie de dimensión 3 dotados de una métrica invariante a izquierda. Estos espacios ambiente generalizan las geometrías de Thurston (salvo S^2 x R) que vienen siendo estudiadas en los últimos años. Dedicaremos la primera sesión a clasificar estos espacios y a entender sus propiedades más básicas. Las herramientas a desarrollar serán de utilidad para estudiar superficies con CMC en la segunda sesión, en la que prestaremos especial atención a los problemas de Hopf y Alexandrov en dichos ambientes.

Gluing is a well established procedure to construct examples of minimal surfaces. In the past year I have been interested in developping tools to glue infinitely many minimal surfaces together. In this talk I will describe several families of minimal surfaces of theoretical interest that were constructed along the way. Then I will explain some of the technicalities involved in this kind of project, including a connection with some discrete analysis problems on graphs.

Recent results allowed to generalized the classical Weierstrass representation of Riemannian surfaces to pluriminimal immersions of Kaehlerian manifold of arbitrary dimension. We show that it is possible to give Weierstrass representations in the very general context of pluri-conformal complex manifolds in arbitrary signature. Using moreover para-complex forms, we find analogous result for pseudo-Riemannian manifolds of split signature (p,p) and necessary and sufficient conditions on these forms for the immersion to be (para-)Kaehler and pluri-minimal.
It is worth pointing out that this framework is particular well adapted to the study of Lorentzian surfaces, which are naturally endowed with a para-complex structure. We consider then existence results for associated families, i.e one-parameter families of (para-)Kaehlerian immersions and give some examples.

Clasificaremos las soluciones a la ecuación de Liouville en el semiplano superior con una cierta condición de Neumann sobre el eje real que genera singularidades. Concretamente, este problema describe métricas conformes de curvatura constante tales que el borde tiene curvatura geodésica constante (respectivamente c₁ en R⁺ y c₂ en R^-). Trabajo conjunto con J.A. Gálvez y Pablo Mira.

In arXiv:1010.1879 we construct an integer valued degree theory for compact, immersed hypersurfaces of constant curvature in general Riemannian manifolds, and we show how this degree theory may be used to prove existence results such hypersurfaces. Joint work with Harold Rosenberg.

We show how barrier techniques are used to solve the Plateau problem for general curvature functions in general manifolds.
arXiv:1008.3545

I will discuss doubling constructions for the Clifford torus and the equatorial two-sphere in the round three-sphere. In doubling constructions minimal surfaces are constructed resembling two copies of the given minimal surface joined by many small catenoidal bridges. I will then discuss desingularization constructions where minimal surfaces are constructed by replacing the intersection curves of minimal two-surfaces in a Riemannian three-manifold with handles modeled after the singly periodic Scherk surfaces and then perturbing to minimality. Finally I will discuss some applications and open questions for closed embedded minimal surfaces in the three-sphere.

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem was open for a long time.
We use an affine approach to this problem. In [1], it was proved by a direct method that in dimension 2, any homogeneous affine manifold admits a homogeneous geodesic. In [2], it was proved by a differential-topological method that any homogeneous affine manifold admits a homogeneous geodesic through each point. As a corollary, we get the same result for homogeneous pseudo-Riemannian manifolds, either reductive or not.
[1] Dusek, Z., Kowalski, O., Vlasek, Z.: Homogeneous geodesics in homogeneous affine manifolds, Results in Math. (2009).
[2] Dusek, Z.: The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds, J. Geom. Phys. (2010).

We first show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid. Second, we prove that an embedded minimal annulus, which lies between two different concentric spheres and meets spheres perpendicularly along its boundaries, is part of a plane.

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $varphi$ be a positive smooth function on $M$. In the warped product $M times_varphi mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.
Joint work with A. Borisenko

I will talk about recent interest to Einstein-Finsler gravity following some quantum gravity phenomenology, exact solutions and Ricci-Finsler flow theory. A survey oriented to both physicists and mathematicians will be presented.

A tensor T is called isotropic if T(v,...,v) is independent of the choice of unit length vector v. The first result about Lagrangian submanifolds admitting an isotropic tensor was due to Naitoh who classified isotropic parallel Lagrangian submanifolds of complex space forms. Later results are due to Montiel, Urbano and Ejiri. In both cases the tensor T(X,Y,Z,W)=< h(X,Y),h(Z,W) >, where h is the second fundamental form of the Lagrangian immersion. The classification result in this case either gives the parallel hypersurfaces of Naitoh or a special class of H umbilical Lagrangian submanifolds. Of course the same question can also be asked for other geometric tensors like T(X,Y,Z,W)=<∇h(X,Y,Z),JW> or T(X,Y,Z,W,U,V)= <∇h(X,Y,Z),∇h (W,U,V)>. The first condition actually can be used to characterize the Whitney spheres, together with the parallel Lagrangian submanifolds. Whereas the second one is more difficult to treat and so far a classification of it is only known in dimension 3. Of course another possibility to generalize the previous results is to look at Lagrangian submanifolds of inefinite complex space forms. As a basic ingredient in the positive definite case which is the choice of a canonical frame based on the choice of e_1 as a vector on which a certain function on a compact set attains an absolute maximum breaks down in the indefinite case; new methods need to be developed. Moreover, the above developped techniques can also be used to study some submanifolds in affine differential geometry. The results in these lectures are based on work in progress with F. Dillen, H.Li and X. Wang for Lagrangian submanifolds and with O.Birembaux and M. Djoric for affine differential geometry.

Para cada k ≥ 1, construimos superficies mínimas propiamente embebidas en H²xR con género cero, infinitos finales asintóticos a planos geodésicos verticales y k finales límite. Dichas superficies se obtienen por conjugación a partir de ciertos grafos de Jenkins-Serrin en H²xR.

Since the harmonicity of the conformal Gauss map characterizes Willmore surfaces, one can use harmonic map methods to obtain results for Willmore surfaces. In particular, we show that a Willmore torus with non-trivial normal bundle comes from holomorphic data by using an analogue of the delbar-sequence for harmonic maps into complex projective space. Moreover, the harmonic conformal Gauss map of a Willmore surface gives an associated family of flat connections, and thus allows to introduce a spectral parameter and to define a spectral curve. We will discuss both of these constructions, and the associated geometric transformations on Willmore surfaces.

Las soluciones de viscosidad de las ecuaciones de Hamilton-Jacobi admiten una interpretación geométrica como la función distancia a la frontera en una variedad de Finsler. En esta interpretación, el lugar singular de las soluciones es el cut locus desde la frontera. Recopilamos resultados sobre el lugar singular usando los dos puntos de vista. A continuación, estudiamos el lugar singular clasificando en distintas categorías todos sus puntos, excepto un conjunto de codimensión de Haussdorff 3. Finalmente, nos preguntamos si es posible caracterizar el lugar singular por dos propiedades del lugar singular, que expresamos diciendo que el lugar singular es un split locus, y que es balanced.

In the first part of this talk we will show some lower estimates for the first eigenvalue of a minimal hypersurface immersed in H^mxR. As an application we prove that if a complete minimal surface in H^2xR has finite total extrinsic curvature, then such surface has finite Jacobi index. In the second part we will apply the method of Colding-Minicozzi to provide some upper estimates for the first eigenvalue of a complete Riemannian surfaces. We then apply this estimates for minimal surface in H3 and H^2xR. These results are part of a joint work with P. Bérard and Ph. Castillon.

We shall see how to prove isoperimetric inequalities on submanifolds of the Euclidean space, using mass transportation methods. We obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality. The proof relies on the description of a solution of the problem of Monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension.

I will go over recent work with Tinaglia on curvature estimates for H-surfaces (constant mean curvature surfaces) in homogeneous 3-manifolds and the classification of CMC-foliations of $R^3$. Next I will go over my recent proof that for each $H>0$, there exists a unique sphere $S_H$ in Sol3 with constant mean curvature H, which is based on previous work of Daniel and Mira.

In this talk I will give and prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to my knowledge its proof has never been published. This proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces.

En espacio-tiempos de Robertson-Walker generalizados 3-dimensionales, estudiamos superficies espaciales completas de curvatura media constante (CMC) acotadas en dos sentidos distintos. En primer lugar, consideraremos superficies comprendidas entre dos slices, y en segundo lugar superficies con ángulo hiperbólico acotado (noción que generaliza la de imagen hiperbólica acotada en el espacio de Lorentz-Minkowski de dimensión 3). En ambos casos obtenemos resultados de no existencia y unicidad, que combinados permiten obtener todas las soluciones enteras y acotadas de la ecuación de superficies CMC en los espacio-tiempos en los que estamos trabajando.

I will present the deformation of Kilian-Schmidt in the context of minimal surfaces of SxR. The annuli remain embedded along the deformation. This work is related to the proof of the uniqueness of Clifford torus in 𝕊³ and Riemann examples in ℝ³.

We discuss the problem of classifying the asymptotic geometry of complete, properly embedded minimal surfaces in ℝ³ with finite topology. The techniques turn out to be dramatically different depending on whether there is one end or more than one. We focus on the former, as it has proven substantially more challenging - requiring a new theory due to Colding and Minicozzi. We outline their work and describe how it is used to prove the result - due to Meeks and Rosenberg - that the plane and helicoid are the only embedded minimal disks. In the process, we indicate how to generalize the argument to positive genus. Indeed, we conclude that such surfaces are asymptotic to a helicoid; justifying the name "genus-g helicoids". This is joint work with C. Breiner.

Vamos a comentar algunos resultados de la teoría de superficies llanas en el 3-espacio hiperbólico, relacionados con nuestra representación conforme. En particular, como los ejemplos de revolución son los únicos completos con una singularidad aislada, mostraremos cómo construir y caracterizar nuevos ejemplos con dos singularidades.

Una variedad riemanniana no compacta es hiperbólica si admite una función superarmónica positiva no constante. En caso contrario, decimos que es parabólica. La hiperbolicidad de una variedad riemanniana se puede caracterizar en términos de la capacidad de la siguiente forma: una variedad es hiperbólica si existe un compacto con capacidad positiva. En esta charla, demostraremos cotas para la capacidad de cuerpos convexos con curvatura media acotada en variedades cuyas curvaturas seccionales (o curvatura de Ricci) están también acotadas.

Se demuestra la existencia de dominios no compactos y no triviales del espacio euclideo para los cuales la derivada normal de la primera función propia del Laplaciano (con condición de Dirichlet cero en el borde) es constante al borde. Estos dominios son obtenidos por perturbación de un cilindro y dan un contraejemplo a una conjectura de Berestycki-Caffarelli-Nirenberg.

Actualmente, el borde conforme es el borde más usado en relatividad matemática. Sin embargo, diversos estudios han demostrado que dicha construcción tiene importantes limitaciones. Como consecuencia, el borde causal, cuya definición ha quedado totalmente justificada recientemente, aparece como una alternativa óptima. El seminario tiene los siguientes objetivos. Primero, explicar brevemente la construcción de ambos bordes. Segundo, ver qué condiciones de regularidad deben imponerse para asegurar que ambas construcciones coincidan, no sólo como conjuntos de puntos, sino como estructuras topológica y cronológica. Finalmente, destacaremos las consecuencias más importantes de este estudio.

Dado D un dominio en una 3-variedad y M una superficie abierta, se dice que D es un dominio de Calabi-Yau para M si no se puede construir una inmersión completa y propia de M en D que tenga curvatura media acotada. Nosotros veremos que si M tiene un final anular, entonces toda tres variedad admite un dominio de Calabi-Yau para M.