I will discuss regularity of solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. The work of E. Giusti and others, establishes a very general existence theory of solutions with "unattained Dirichlet data" by minimizing an appropriately defined functional, which includes information about the boundary data. We can naturally associate to such a solution a current, which inherits a natural minimizing property. The main goal is to show that its support is a $C^{1,alpha}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class $C^{1,alpha}$. Furthermore, as a consequence, I will discuss some interesting results about the trace of such a solution; in particular for a large class of boundary data with jump discontinuities, the trace has a jump discontinuity along which it attaches to the vertical part of the prescribed boundary.

We present two recent results on $r$-minimal hypersurfaces in Euclidean space, namely, an existence result for $r$-minimal hypersurfaces with ends of planar type via perturbative methods (joint work with J. de Lira and J. Silva) and a uniquess result, a la Schoen, for a certain class of two-ended $r$-minimal hypersurfaces (joint work with A. Souza).

We define closed partially conformal vector fields and use them to give a characterization of Riemannian manifolds which admit this kind of fields as some special warped products foliated by (n-1)-umbilical hypersurfaces. In the particular case of Euclidean spaces, the foliations are then given by hyperplanes, hyperspheres or by coaxial cylinders. As an application, we obtain conditions for a constant mean curvature hypersurface to be (n-1)-umbilical.

Sea M una superficie de Hadamard con curvatura de Gauss acotada por arriba por una constante negativa. Resolvemos el problema de existencia de difeomorfismos armónicos desde el plano complejo en M. Para ello, estudiaremos grafos mínimos sobre dominios convexos no acotados de MxR. Presentaremos un teorema tipo Jenkins-Serrin en MxR y mostraremos la existencia de grafos mínimos enteros con el tipo conforme del plano complejo.

El objetivo de la charla es resumir algunos temas de Geometría Lorentziana actuales y que se están estudiando en Granada y grupos afines, concretamente: Técnicas variacionales y topológicas sobre geodésicas: conectividad geodésica Problemas específicos de espaciotiempos estacionarios: estructura, métricas de Fermat. Bordes causales: el problema de su definición, relevancia en la correspondencia AdS/CFT. De cada uno se explicará el interés del problema y las técnicas que se usan.

Se va a plantear y resolver el problema tipo Björling de determinar la existencia y unicidad de superficies maximales afines conteniendo a una curva y con un normal afín preescrito a lo largo de ella. También se aplicarán los resultados al estudio de simetrías, geodésicas y singularidades en superficies maximales afines.

Tranformaciones de Ribaucour para superficies en el 3-espacio Euclídeo, parametrizadas por líneas de curvatura fueron clásicamente estudiadas por Bianchi. Fueron estudiadas transformaciones de Ribaucour entre superficies mínimas y de curvatura de Gauss constante. Presentaremos ejemplos de superficies mínimas y de curvatura media constante obtenidas por transformaciones de Ribaucour. Presentamos transformaciones de Ribaucour entre superficies de Weingarten lineales. Tambien presentamos datos holomorfos de superficies de Weingarten Generalizadas de tipo Bryant en el 3-espacio hiperbolico que incluyen las superfices de Bryant y las superficies Llanas.

Este minicurso va dirigido a estudiantes de posgrado que hayan completado al menos un primer curso de geometría diferencial. Nuestro objetivo es introducir a los alumnos en algunas de las técnicas del análisis geométrico y en sus aplicaciones al estudio de la geometría global de hipersuperficies minimales en la esfera. En particular, nos proponemos hacer accesibles al nivel de los estudiantes distintos resultados de investigación, más o menos recientes, sobre el índice y la estabilidad de tales hipersuperficies. Este minicurso está basado en el trabajo titulado "On the stability index of minimal and constant mean curvature hypersurfaces in spheres", publicado por el autor en la Revista de la Unión Matemática Argentina y disponible en http://inmabb.criba.edu.ar/revuma/pdf/47-2/alias.pdf

We consider operator L acting on functions on a Riemannian surface of the form $L = Delta + V - aK$. Here $Delta$ is the Laplacian, $V$ is a non-negative potential, $K$ is the Gaussian curvature and $a$ is a non-negative constant. Such operators $L$ arise as the stability operator of immersed in a Riemannian 3-manifold with constant mean curvature (for particular choices of $V$ and $a$). We assume $L$ is non-positive acting on functions compactly supported and we obtain results in the spirit of some theorems of Ficher-Colbrie-Shoen, Colding-Minicozzi and Castillon. We extend these theorems to $a leq 1/4$

We construct complete minimal surfaces with two ends in Euclidean 3-space. In particular, we prove the existence of such surfaces with arbitrary even number of genus which minimize the degree of the Gauss map for their genus. We introduce some remaining problems as well.

We will show that the index of a lightlike geodesic in a standard stationary spacetime $(mathcal M_0 imesR,g)$ is equal to the index of its spatial projection as a geodesic of a Finsler metric $F$ on $mathcal M_0$ associated to $(mathcal M_0 imesR,g)$. Moreover we obtain the Morse relations of lightlike geodesics connecting a point $p$ to a curve $gamma(s)=(q_0,s)$ by using Morse theory on the Finsler manifold $(mathcal M_0,F)$. Finally, we will show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.

This is a joint work of Kokubu, Rossman, Umehara and Yamada. We talk on the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space $H^3$. Gálvez, Martínez and Milán showed that when the singular set does not accumulate at an end, then the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called pitch p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have $-1 < ple 0$. If the singular set accumulates at the end, the pitch p is a positive rational number not equal to 1. Choosing appropriate positive integers n and m so that $p=n/m$, suitable slices of the end by horospheres are asymptotic to epicycloids or hypocycloids with 2n cusps and whose normal directions have winding number m. Furthermore, it is known that the caustics of flat surfaces are also flat. So, as an application, we give a useful explicit formula for the pitch of ends of caustics of complete flat fronts.

Roughly speaking, it is expected that the only two types of singular laminations that can occur as limits of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-properhelicoid-like limits. T. Colding and C. De Lellis constructed an example of the first type. I will present a construction of the second type: we show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal tori that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. This is a joint work with Darren Lee.

This is a joint work of Saji, Umehara and Yamada. We talk on the recognition problem of Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role to generalize the authors previous work the geometry of fronts for surfaces. The crucial point to prove our criteria for Ak-singularities is to introduce a suitable parametrization of the singularities called the k-th KRSUY-coordinates. Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a hypersurface (as a wave front) in $R^{n+1}$ is differentiable right-left equivalent to the $A_{k+1}$-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane $R^n$ is right-left equivalent to the $A_k$-type singular point in $R^n$. Moreover, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.

Conferencia dentro del programa de postgrado Máster de Matemáticas.

En principio, las superficies minimales completas parecen estar alejadas del significado original de superficie minimal, esto es, soluciones del problema de Plateau. El estudio de las superficies minimales completas y acotadas ha establecido una sorprendente relación entre ellas y el problema de Plateau: la existencia de superficies minimales completas y con borde en R3. Este tipo de superficies son muy numerosas: existen ciertos teoremas de densidad. Por otro lado, condiciones sobre el conjunto límite dan lugar a teoremas de no existencia.

We show the existence in $H^2xR$ of a family of surfaces of genus $k > 0$, finite total extrinsic curvature with two catenoidal ends and one middle planar end. The proof is based on a gluing procedure which involves a compact part of a scaled Costa-Hoffman Meeks surface of $R^3$, two pieces of a catenoid of $H^2xR$ and the hyperbolic plane $H^2x{0}$ from which we have removed a disk.

The 3-dimensional Lie group $\widetilde{PSL_2(R)}$ is a Riemannian fiberation over the hyperbolic plane, with a 4-dimensional isometry group. We give a description of this isometry group and describe the minimal surfaces invariant under 1-parameter subgroups. We then establish sufficient machinary to show a Jenkins-Serrin type theorem in this space.

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.

En este trabajo, analizamos algunos aspectos de la distancia lorentziana en espacio-tiempos con curvatura seccional (o de Ricci) controlada. En particular, estudiamos la restricción de dicha distancia a hipersuperficies espaciales con curvatura de Ricci con decrecimiento cuadrático fuerte. Como consecuencia, y bajo hipótesis adecuadas sobre las curvaturas (seccionales o de Ricci) del espacio-tiempo ambiente, obtenemos estimaciones para la curvatura media de dichas hipersuperficies y damos una condición suficiente para su hiperbolicidad. Trabajo conjunto con L. J. Alias y V. Palmer.

The focus of this work is on a conjecture of Blaine Lawson that the only embedded minimal 2-torus in the standard 3-sphere is the Clifford torus. Our approach to resolving this conjecture is to view the larger class of minimal immersions of the plane in terms of the nullity of the immersions, the kernel of the second variation of the area functional. We examine the case where the nullity is restricted. This restriction, conveniently, allows us to construct a dimension-2 foliation (for each set of initial conditions) on $SO(4)xR^2$, the leaves of which project to minimal immersions of the plane into the sphere. We proceed to show that all compact images, all minimal immersions of the torus with restricted nullity, are known examples, the only ebedded one of which is the Clifford torus. Thus any counter-example to this conjecture would have to have large nullity. Joint work with Oscar Perdomo (Universidad del Valle/Central CT State U.)

In this talk we will speak about a forgotten proof due to René Garnier (1928) of Plateau problem in the 3 dimensional euclidean space, in the case of polygonal boundary. The proof is based on the fact that we can associate a Fuchsian equation to any minimal surface bounded by a polygon. The monodromy of the equation is given by the directions of the edges of the polygon. To solve Plateau problem, we are thus led to solve a Riemann-Hilbert problem and to construct isomonodromic deformations. In the case of a quadrilateral boundary, this deformations are given by the Painlevé VI equation.

En esta charla abordamos el problema de clasificar los grafos enteros minimales dentro del espacio de Heisenberg riemanniano 3-dimensional $\mathrm{Nil}_3$. Veremos cómo el espacio de tales grafos enteros minimales es una clase amplísima, y describiremos una parametrización para él en términos de diferenciales cuadráticas holomorfas definidas en el plano complejo o en el disco unidad. La clave para dicha resolución del problema de Bernstein en $\mathrm{Nil}_3$ reside en la construcción previa de una aplicación de Gauss armónica sobre el disco de Poincaré para la teoría "hermana" de superficies de curvatura media constante $H=1/2$ dentro del espacio producto $H^2\times \mathbb{R}$.

We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of Laplace-Beltrami operator in some Riemannian manifold. These domains are close to geodesic spheres of small radius centered at a nondegenerate critical point of the scalar curvature.

Dado un espaciotiempo estacionario estándar, las geodésicas luminosas se proyectan salvo reparametrización sobre geodésicas de una métrica de Finsler que llamamos "métrica de Fermat". Usando esta relación, demostramos diversos resultados de existencia y multiplicidad de rayos de luz, a través de la geometría de Finsler. Finalmente consideramos también las geodésicas temporales usando un truco de tipo Kaluza-Klein.

Conferencia dentro del programa de postgrado Máster de Matemáticas

Conferencia dentro del programa de postgrado Máster de Matemáticas

We prove existence of graphs over exterior domains of $H^2$ with constant mean curvature $H=1/2$ in $H^2xR$, provided the boundary curve satisfies some geometric conditions. Furthermore, we prove a vertical halfspace theorem for surfaces with constant mean curvature $H=1/2$ properly immersed in $H^2xR$.

A classical geometrical interpretation for the squared length of the second fundamental form, or Casorati curvature, of a surface in Euclidean three-space will be recalled and extended for submanifolds with arbitrary codimension in a Riemannian manifold. We will argue that the Casorati curvature coincides with our "intuitive" notion of curvature. This curvature is then used to obtain an optimal inequality between the intrinsic scalar curvature and an extrinsic curvature invariant of a submanifold in a Riemannian space form. The equality case in the inequality characterizes totally quasi-umbilical submanifolds.

En este trabajo conjunto con Laurent Mazet y Harold Rosenberg, encontramos condiciones necesarias y suficientes para la existencia de solución del problema de Dirichlet en dominios de tipo Jenkins-Serrin generales de H2xR para valores frontera continuos a trozos arbitrarios (posiblemente $pminfty$), y estudiamos la unicidad de dicha solución. Estos dominios no son necesariamente simplemente conexos ni acotados, y su frontera está formada por un número finito de curvas convexas respecto al dominio.

Motivated by the dynamics theorem for minimal surfaces by Meeks, Perez and Ros, then Meeks and Tinaglia proved a related result for CMC surfaces in $\mathbb{R}^3$. This theorem has important applications to the study of complete embedded CMC surfaces of finite topology in complete locally homogeneous three-manifolds. I will only be discussing the statements and proofs of the CMC dynamics theorem in $\mathbb{R}^3$, rather than applications.