A semi-Riemannian manifold is geodesically complete (or for short, complete) if its maximal geodesics are defined for all times. For Riemannian metrics the compactness of the manifold implies completeness. In contrast, there Lorentzian metrics on the torus that are not complete. Nevertheless, completeness plays an important role for fundamental geometric questions in Lorentzian geometry such as the classification of compact Lorentzian symmetric spaces and in particular for a Lorentzian version of Bieberbach's theorem. We will study the completeness for compact manifolds that arise from the classification of Lorentzian holonomy groups, which we will briefly review in the talk. These manifolds have abelian holonomy and carry a parallel null vector field. By determining their universal cover we show that they are complete. In the talk we will explain this result and further work in progress, both being joint work with A. Schliebner (Humboldt-University Berlin).

I will introduce Lyapounov functions for cone fields, a generalization of the causal structure of a Lorentzian metric, and present some results on their existence. If time permits I will define a notion of global hyperbolicity for cone fields and give a result on the existence of steep Lyapounov functions for globally hyperbolic cone fields. The material is a cooperation with Patrick Bernard (Paris).

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface \(\Sigma\subset\Omega\) with the simple lift property must have genus zero, if \(\Omega\) is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that \(\Omega\) cannot contain closed minimal surfaces. In this work, I generalize this result by taking an arbitrary orientable three-manifold \(\Omega\) and proving that one is able to restrict the topology of an arbitrary surface \(\Sigma\subset\Omega\) with the simple lift property. Among other things, I prove that the only possible compact surfaces with the simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where \(\Sigma\subset\Omega\) is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.

In \(R^n\) (\(n\) at most 7) the famous Bernstein's theorem states that every entire solution to the minimal graph equation must be affine. Moreover, entire positive solutions in \(R^n\) are constant in every dimension by a result due to Bombieri, De Giorgi and Miranda. If the underlying space is changed from \(R^n\) to a negatively curved Riemannian manifold, the situation is completely different. Namely, if the sectional curvature of \(M\) satisfies suitable bounds, then \(M\) possess a wealth of solutions.
One way to study the existence of entire, continuous, bounded and non-constant solutions, is to solve the asymptotic Dirichlet problem on Cartan-Hadamard manifolds. In this talk I will discuss about recent existence results for minimal graphs and f-minimal graphs. The talk is based on joint works with Jean-Baptiste Casteras and Ilkka Holopainen.

Obtenemos un teorema de caracterización del cilindro construido sobre una curva grim reaper como el único solitón de traslación del flujo por la curvatura media en el espacio euclídeo \(R^n\) que es asintótico a dos hiperplanos fuera de un cilindro. Esto generaliza a dimensión arbitraria un resultado previo de Martin-Perez-Savas-Smoczyk para solitones de \(R^3\).

In the 1950s, Lax and Malgrange proved that a solution v of a linear elliptic equation with analytic coefficients Pv=0 in a compact set can be approximated by a global solution u of Pu=0 provided that the complement of the set is connected. This theorem is key to showing the existence of minimal graphs on the unit ball whose transverse intersection with a horizontal hyperplane has arbitrarily large (n-1)-measure. The proof hinges on the construction of minimal graphs which are almost flat but have small oscillations of prescribed geometry. In addition, I will present an overview of our recent generalization of the Lax-Malgrange result to the case of parabolic equations and discuss applications to the study of hot spots. This is a joint work with A. Enciso and D. Peralta-Salas.

The Heisenberg group has a well studied sub-Riemannian structure. Geometric measure theory in the sub-Riemannian setting is still in development and several fundamental questions are still open. One reason is that sets of finite sub-Riemannian perimeter may have fractal behaviours. I will present the state of art (in my knowledge) of the study of minimal surfaces in this setting, with some recent results about contact variations of the area functional.

In a lesser-known paper [Berl. Ber. 1921, pp. 261-264], Albert Einstein proposed the natural addition of a scalar differential equation to the field equations of General Relativity. I shall explore this suggestion from a geometrical point of view. As a consequence, I will show a family of scalar conformal invariants for generic (pseudo-)Riemannian manifolds of dimension greater than 3.

La demostración de la desigualdad de Riemann-Penrose por parte de Huisken e Ilmanen al final del pasado siglo fue un avance importante tanto para la Teoría de Agujeros Negros como para la Geometría Diferencial. Para la primera, tal desigualdad puede ser vista como el mejor test de su consistencia física, debido a la dificultad en observar evidencias directas de los agujeros negros. Para la Geometría Diferencial, la demostración puso de manifiesto cuán poderoso y versátil es el llamado flujo inverso de la curvatura media, incluyendo la formulación débil de las soluciones, que permite a las superficies del flujo ”saltar” más allá de las singularidades. De hecho, este flujo fue usado, poco después, por Bray y Neves para calcular el invariante de Yamabe del espacio proyectivo real de dimensión 3. El propósito de este curso es el de estudiar en detalle la demostración de Huisken e Ilmanen, proporcionando previamente una perspectiva de las motivaciones físicas y de las herramientas acerca de flujos geométricos que son necesarias para su comprensión.

La demostración de la desigualdad de Riemann-Penrose por parte de Huisken e Ilmanen al final del pasado siglo fue un avance importante tanto para la Teoría de Agujeros Negros como para la Geometría Diferencial. Para la primera, tal desigualdad puede ser vista como el mejor test de su consistencia física, debido a la dificultad en observar evidencias directas de los agujeros negros. Para la Geometría Diferencial, la demostración puso de manifiesto cuán poderoso y versátil es el llamado flujo inverso de la curvatura media, incluyendo la formulación débil de las soluciones, que permite a las superficies del flujo ”saltar” más allá de las singularidades. De hecho, este flujo fue usado, poco después, por Bray y Neves para calcular el invariante de Yamabe del espacio proyectivo real de dimensión 3.
El propósito de este curso es el de estudiar en detalle la demostración de Huisken e Ilmanen, proporcionando previamente una perspectiva de las motivaciones físicas y de las herramientas acerca de flujos geométricos que son necesarias para su comprensión.

In this talk we will present a version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol3. Moreover, we will construct examples of admissible domains where the results may be applied. This is a joint work with Patricia Klaser.

Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S) >= Area(W), provided the intersections of S with horizontal planes have the same orientation.This is joint work with Jaigyoung Choe.

Translating Solitons are some special solutions to the Mean Curvature Flow. The classical examples appear in the Euclidean Space. We review some basic properties, both in Euclidean and Minkoswki Space. Here, we will pay attention to those whose direction of translation is lightlike, in dimensions 2 and 3, obtaining some examples.

In this talk we will present the main results of a work in collaboration with S. Canevari and G. M. de Freitas. More precisely, we prove that complete submanifolds, on which the weak Omori-Yau maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the larger such curvatures are. This work unifies and generalizes several previous results on submanifolds with nonpositive extrinsic curvature.

Less than two years ago some physicists gave the official announcement that gravitational waves exist, but, from a geometrical point of view, they have always been “real objects” and their properties have been widely investigated.
The aim of this talk is introducing generalized plane waves and discussing some of their properties such as geodesic connectedness and geodesic completeness.

En esta charla expondré algunas nociones básicas de la Relatividad Especial, sin apenas fórmulas, a través de sencillos diagramas en el espacio de Minkowski. Usando intuitivos argumentos geométricos, explicaré la paradigmática paradoja de los gemelos, y las menos conocidas paradojas de Bell y de la pértiga y el granero.

The concept of trapped surfaces was originally formulated by Penrose for the case of 2-dimensional spacelike surfaces in 4-dimensional spacetimes in terms of the signs or the vanishing of the so-called null expansions. This is obviously related to the causal orientation of the mean curvature vector of the surface, which provides a better and powerful characterization of the trapped surfaces and allows the generalization of this concept to codimension two spacelike submanifolds of arbitrary dimension n. In this sense, an n-dimensional spacelike submanifold \(\Sigma\) of an (n + 2)-dimensional spacetime is said to be future trapped if its mean curvature vector field \(\vec{H}\) is timelike and future-pointing everywhere on \(\Sigma\), and similarly for past trapped. If \(\vec{H}\) is lightlike (or null) and future-pointing everywhere on \(\Sigma\) then the submanifold is said to be marginally future trapped, and similarly for marginally past trapped. Finally, if \(\vec{H}\) is causal and future-pointing everywhere, the submanifold is said to be weakly future trapped, and similarly for weakly past trapped. The extreme case \(\vec{H} = 0\) on \(\Sigma\) corresponds to a minimal submanifold. In this lecture we consider codimension two compact marginally trapped submanifolds in the light cone of de Sitter space. In particular, we show that they are conformally diffeomorphic to the round sphere and, as an application of the solution of the Yamabe problem on the round sphere, we derive a classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space. This is part of our work in progress with Luis J. Alías (Univ. Murcia) and Marco Rigoli (Univ. Milano).

My talk will begin with a survey on the existence theorem of a neutral metric on a 4-manifold. The existence of a neutral metric on a 4-manifold is equivalent to the existence of a field 2-planes, and also to that of a pair of two kinds of almost complex structures. These facts are proved on the basis of a theorem of Hirzebruch and Hopf (1958), and of Donaldson’s classification of definite intersection forms (1983). I then consider the Goldberg Conjecture (1969) of indefinite metric version, and I report some counterexamples to the Conjecture of indefinite metric version. Also, future problems will be discussed.

In this talk. first, from the histrorical view point we introduce some works on isometric Reeb flow on real hypersurfaces in complex projective space, complex hyperbolic space, complex two-plane Grassmannians, and complex quadrics. Next we want to give a recent classification of isometric Reeb flow on real hypersurfaces in generalized complex k-plane Grassamnnians.

We prove that every meromorphic function on an open Riemann surface \(M\) is the complex Gauss map of a conformal minimal immersion \(f:M\to \mathbb R^3\); furthermore, \(f\) may be chosen as the real part of a holomorphic null curve \(F:M\to\mathbb C^3\). Analogous results are proved for conformal minimal immersions \(M\to\mathbb R^n\) for any \(n>3\). We also show that every conformal minimal immersion \(M\to\mathbb R^n\) is isotopic to a flat one, and we identify the path connected components of the space of all conformal minimal immersions \(M\to\mathbb R^n\) for any \(n\ge 3\). (Joint work with Antonio Alarcón and Francisco J. López,

Lawson correspondence between minimal surfaces in \(\mathbb{S}^3\) and constant mean curvature (CMC) surfaces in \(\mathbb{R}^3\) is instrumental in building examples of surfaces, but also reveals a lot about the structure of underlying PDEs. In discrete geometry, simple definitions of CMC surfaces are not obvious (many have been given) and an analog to Lawson correspondence has eluded researchers for a long time. We will present here a natural and computation-friendly definition, based on Lax pairs, which showcases the analogy between the two types of surfaces, as well as proposes a construction method. We will try to outline the similarities and differences between the smooth and discrete realms. Joint work with Alexander Bobenko (TU Berlin)

The purpose of this article is to provide sufficient conditions for having a positive weak solution for a fourth-order non-linear partial differential equation in M, a compact Riemannian manifold, and this solution is harmonic on the boundary of M.