In this talk, we prove that the Min-Oo's conjecture holds if we consider a compact connected locally conformally flat manifold with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality, and the mean curvature of the boundary is controled bellow by the mean curvature of a geodesic ball in the standard unit-sphere. This is a joint work with E. Barbosa and M.P. Cavalcante.

Director: Francisco Martín.

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

Manifolds and spaces of non-positive curvature
Ideal boundary
Contracting boundary
Applications to foliations and laminations.

We will study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds \(X\) that can be expressed as a semidirect product of \(\mathbb{R}^2\) with \(\mathbb{R}\) endowed with a left invariant metric. For any such compact minimal surface \(M\), we provide an a priori radius estimate which depends only on the maximum distance of points of the boundary \(\partial M\) to a vertical geodesic of \(X\). In particular, there are no complete stable minimal surfaces inside solid metric cylinders around vertical geodesics in \(X\) .​ We also give a generalization of the classical Rado's Theorem to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in \(X\) , and we study the geometry, existence and uniqueness of this type of Plateau​ problem.​ Joint work with Bill Meeks and Pablo Mira.​

In my talk I explain how Quaternionic Holomorphic Geometry can be used to discuss the associated family of isothermic surfaces. Isothermic surfaces are surfaces which have a conformal curvature line parametrisation, and cylinder, minimal surfaces and constant mean curvature surfaces are examples. It is known that any isothermic surface has a family of flat real connections such that parallel sections give new isothermic surfaces. In this talk, I will introduce a complex family which specialises to the known complex families for minimal surfaces and constant mean curvature surfaces.

This talk presents an algebraic invariant that measures the degree to which a (non-standard) static or stationary spacetime can exhibit violations of naive expectations on causal structure; for a static spacetime, this invariant ("fundamental cocycle") is a de Rham cohomology class, but for stationary spacetimes it is something of looser structure.  Using this cocycle provides a simple measure for determining whether two events are timelike related, so it leads to classification of a spacetime along the "causal ladder" (globally hyperbolic, strongly causal, etc.).  It has excellent properties with group actions, leading to conclusions about when spacetimes and their covers or quotients have similar causality properties.  In particular, it can be used to create (by means of quotient) spacetimes of desired causality structure, such as cosmic strings on non-flat backgrounds.

In the last years a great deal of e ort has been invested in order to investigate the existence of geodesics connecting two given points in Lorentzian manifolds as a statement similar to the Hopf-Rinow Theorem is not known for spacetimes.
In particular, some geometric sufficient conditions have been introduced so that stationary spacetimes, i.e. spacetimes equipped with a timelike Killing vector eld, are geodesically connected. Unluckily, a similar global result is not true for space- times equipped with a lightlike Killing vector eld. Anyway, points connected by geodesics can be characterized.
Joint works with Rossella Bartolo and José Luis Flores.

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

I will present a general method (pioneered by Ros and Savo) to obtain universal and effective index estimates for minimal hypersurfaces inside a Riemannian manifold, given an isometric embedding of the latter in some (possibly high-dimensional) Euclidean space. This approach can be applied, on the one hand, to tackle a conjecture by Schoen and Marques-Neves asserting that the Morse index of a closed minimal hypersurface in a manifold of positive Ricci curvature is bounded from below by a linear function of its first Betti number, which we settle for a large class of ambient spaces. On the other hand, these methods turn out to be very powerful in studying free-boundary minimal hypersurfaces in Euclidean domains: among other things, we prove a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Applications to compactness theorems, to the explicit analysis of known examples (due to Fraser-Schoen and to Folha-Pecard-Zolotareva) and to novel classification theorems will also be mentioned. This is joint work with Lucas Ambrozio and Benjamin Sharp.

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

Abstract: We present some basic notions on magnetic curves on Riemannian manifolds and give several examples in dimension 3, emphasizing the case of Killing magnetic curves. We present some results on magnetic curves in almost contact metric geometry in arbitrary dimension. Later on we introduce the notion of magnetic map between Riemannian manifolds. Magnetic maps are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of them. Further on we describe the problem in almost contact metric geometry. Then we produce examples of magnetic maps, having as either source or target manifold the tangent bundle of a Riemannian manifold equipped with several Riemannian metrics. In particular we study when the canonical projection, a vector field and the tangent map are, respectively, magnetic maps.

Temas:

• Magnetic curves on Riemannian manifolds
• Geodesics and harmonic maps
• Magnetic maps: definition and first examples
• Magnetic maps in almost contact metric geometry
• Magnetic maps and tangent bundle of a Riemannian manifold

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.

We introduce a quaternion invariant for an inclusive immersion in a quaternion manifold, which is a quaternion object corresponding to the Willmore functional. The lower bound of this invariant is given by topological one and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternion projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternion invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point of the invariant under any variations such that the induced complex structures do not vary.

The Brunn-Minkowski inequality can be summarized by stating that the volume, i.e., the Lebesgue measure in \(\mathbb {R}^n\), is (\(1/n\))-concave. More precisely, we have \begin{equation}(1)\quad vol\big((1-\lambda)A+\lambda B\big)^{1/n}\geq(1-\lambda)vol(A)^{1/n}+\lambda vol(B)^{1/n} \end{equation} for all measurable sets \(A, B\) so that \((1-\lambda) A+\lambda B\) is also measurable. It is easy to see that (1) is also true if we exchange \(1/n\) by an arbitrary \(p\leq 1/n\). When working with absolutely continuous measures \(d\mu(x)=f(x)dx\) associated to densities \(f\) with some convexity assumptions, one can also obtain the following Brunn-Minkowski inequality \begin{equation}(2)\quad \mu((1-\lambda) A+\lambda B)^{p}\geq(1-\lambda)\mu(A)^p+\lambda\mu(B)^p \end{equation} for any pair of measurable sets \(A, B\) with \(\mu(A)\mu(B)>0\) and such that \((1-\lambda) A+\lambda B\) is also measurable, where \(p\leq 1/n\) is associated to the ``type of convexity'' of \(f\). The convexity conditions of such density functions \(\)f allow us to understand whether the volume should be the sole measure satisfying the latter inequality or not. Thus, in this talk we will discuss whether, for a given measure on \(\mathbb {R}^n\) (not necessarily absolutely continuous), having an inequality like (2) for a certain (`small') subfamily of sets in \(\mathbb {R}^n\) implies that the measure is (up to a constant) the volume itself. To this respect, we will point out that the constraints \(1/n\geq p>0\), when the support of the measure is the whole \(\mathbb {R}^n\), and \(p=1/n\), when it is an arbitrary open convex set, are both necessary in order to get such a characterization.

Es conocido que las superficies mínimas en \(\mathbb{R}^3\) y las llanas de \(\mathbb{H}^3\) admiten representaciones homomorfas y comparten varios e interesantes aspectos tanto de tipo geométrico como tipológico. A pesar de esto, no se ha dado hasta ahora ninguna descripción geométrica que conecte estos dos tipos de superficies inmersas en diferentes espacios ambiente. Nuestro objetivo es mostrar una construcción geométrica que asocia toda superficie llana de \(\mathbb{H}^3\) (con singularidades admisibles) a un par de mínimas en \(\mathbb{R}^3\) relacionadas por una transformación de Ribaucour. La construcción es un caso particular de una conexión geométrica general entre superficies del espacio hiperbólico y envolventes a una congruencia de esferas del espacio euclídeo.

La conjetura de Hoffman-Meeks afirma que si M es una superficie mínima con curvatura total finita en ​​\(\mathbb{R}^3\)​ con género \(g\) y \(k\) finales, entonces \(k\leq g+2\). Este problema abierto motiva estudiar los posibles límites de una sucesión de superficies mínimas embebidas \(M_n\subset \mathbb{R}^3\)​ con género fijo y curvatura total finita. Normalmente, los objetos que aparecen en el límite son laminaciones mínimas con singularidades. Mediante el uso de la teoría de Colding-Minicozzi, daremos un resultado de convergencia para una parcial de la sucesión \(M_n\) anterior, si asumimos una cota uniforme del radio de inyectividad de las superficies \(M_n\) fuera de un cerrado numerable de \(\mathbb{R}^3\). Usaremos este resultado de convergencia para obtener una cota (no explícita) \(k\leq C(g)\) del número de finales en la conjetura de Hoffman-Meeks, que sólo depende del género. Esta cota del número de finales produce una cota para el índice de estabilidad de una superficie mínima de curvatura total finita, también en función sólo de su género.​

Un famoso teorema de Hopf establece que toda esfera de curvatura media constante en \(\mathbb{R}^3\) es una esfera redonda. En esta charla generalizaremos el teorema a clases de superficies gobernadas por EDPs elípticas sobre cada plano tangente en 3-variedades arbitrarias, con la única hipótesis de la existencia de una familia transitiva de superficies candidatas. Así, probamos que toda esfera en estas condiciones es una esfera de la familia de candidatos. Como aplicación, demostraremos una conjetura de 1956 de A.D. Alexandrov sobre la unicidad de esferas de curvaturas predeterminadas en \(\mathbb{R}^3\) , y completaremos la caracterización de las esferas redondas como las únicas esferas de Weingarten elípticas en \(\mathbb{R}^3\). Este es un trabajo con José A. Gálvez.

Let \(M\) be an open Riemann surface. It was proved by Alarcón and Forstneric that every conformal minimal immersion \(M\to\mathbb R^3\) is isotopic to the real part of a holomorphic null curve \(M\to\mathbb C^3\). We prove the following substantially stronger result in this direction: for any \(n\ge 3\), the inclusion of the space of real parts of non flat null holomorphic immersions \(M\to\mathbb C^n\) into the space of non flat conformal minimal immersions \(M\to \mathbb R^n\) satisfies the parametric h-principle with approximation; in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps. For an open Riemann surface \(M\) of finite topological type, we obtain optimal results by showing that the above inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences. (Joint work with Finnur Lárusson.)

I will talk about the existence and the non existence of minimal graphs in the Heisenberg space, with a given boundary. Moreover height and area estimates for such graphs are obtained.

The talk gives an overview over some recently developed methods in global analysis and geometry that involve conformal factors. First we review a global existence result, obtained with Nicolas Ginoux, for Dirac-Higgs-Yang-Mills systems under the assumption that the underlying spacetime has a conformal extension, which ist the case for solutions to the Einstein equations for initial values in a weighted neighborhood of the standard ones. Then we switch to Riemannian geometry and show, using the novel ‚flatzoomer’ method, that every conformal class contains a metric of bounded geometry. Finally we sketch the consequences of the result for the Yamabe flow on noncompact manifolds and a related result for Cheeger-Gromov convergence of some relevance in the context of positive scalar curvature on compact manifolds.

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and we give new examples of ZMC surfaces containing lightlike lines and a zero mean curvature entire graph of mixed type.

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field \(T\), that is, \(R_{\xi}\phi T=TR_{\xi}\phi\), where \(T=A\) or \(T=S\) for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators, We give a complete classification of real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting condition respectively.

Finsler-Lagrange geometries are a generalization of Riemannian geometry, obtained by allowing the metric tensor to depend not only on the points of the manifold under discussion, but also on a tangent vector at each of these points. We present here in brief the specific features of these geometries together with some of their applications - with a special focus on classical field theories - and the surrounding open questions.

Defining a relevant notion of constant mean curvature and constant mean curvature surfaces is difficult in discrete geometry. In order to make sense of these definitions, we will first recall what properties are shared by the smooth CMC surfaces, e.g. criticality, associated family, integrable system PDE, that we would like to hold also in the discrete case. Then we will describe how such notions can or cannot be carried out in the discrete case. (Joint work with Sasha Bobenko)

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

Topics: Introduction and basic concepts Discrete curves in the plane. Aproximation properties Discrete surfaces. Mean curvature flow. Discrete Gauss-Bonnet theorem

Introducción y conceptos básicos. Primera fórmula de variación y aplicaciones. Teorema de Nitsche. Aplicaciones conformes. Teorema de Fraser-Schoen. Segunda fórmula de variación y aplicaciones. Teorema de Shiffman.

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were 2-dimensional surfaces in the 3-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by De Giorgi in the case of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others). In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in \(\mathbb C^n\) is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. In this talk I will try to give a feeling for the difficulties in the proof and how they can be overcome. Moreover I will touch some recent developments which go beyond Almgren's result.