In the theory of general relativity, spacetimes are modelled by differential manifolds endowed with a Lorentzian metric of regularity (at least) $C^2$. In recent years, there has been a growing interest in the study of spacetimes with Lorentzian metrics of lower regularity (see [1] and [2]), however, some classical and important results on the geometry of spacetimes are no longer valid in this context. Recently, M. Kuzinger and C. Sämann obtained a synthetic approximation to the Lorentzian geometry and causal theory, the so called (pre)-Lorentzian length spaces, which gives an analogous theory to the one of length spaces in the context of Lorentzian geometry (see [3]). In this talk we will show some results concerning to the causal ladder for Lorentzian length spaces.

[1] P. T. Chrusciel and J.D.E. Grant. On Lorentzian causality with continuous metrics, Class. Quantum Grav. 29 145001 (2012).

[2] M. Graf, J.D.E. Grant, M. Kuzinger and R. Steinbauer. The Hawking-Penrose singularity theorems for $C^{1,1}$-Lorentzian metrics, Commun. Math. Phys. 360: 1009. (2018).

[3] M. Kuzinger and C. Sämman. Lorentzian length spaces. C. Ann Glob Anal Geom (2018) 54: 399. https://doi.org/10.1007/s10455-018-9633-1.

The so called Principal Curvature Theorem (PCT) is a purely geometric result on the principal curvatures of complete hypersurfaces in Euclidean space given by Smyth and Xavier (Invent. Math. 90:443--450, 1987) in their proof of Efimov's theorem in dimension greater than two. As another application of the PCT, they also proved that the only complete hypersurfaces immersed in $\mathbb{R}^{n+1}$ with constant mean curvature $H\neq 0$ and having non-positive Ricci curvature are the right circular cylinders of the form $\mathbb{R}^{n-1}\times\mathbb{S}^1(r)$, with $r>0$, extending to the $n$-dimensional case a previous result for $n=2$ due to Klotz and Osserman.

In this lecture we will introduce new applications of the PCT to the study of complete hypersurfaces with constant mean curvature immersed into the Euclidean space $\mathbb{R}^{n+1}$, and, more generally, with constant higher order mean curvature. For instance, among other results, we will prove that if $M^n$ is a complete hypersurface in $\mathbb{R}^{n+1}$ ($n\geq 3$) with constant mean curvature $H\neq 0$ and having two distinct principal curvatures, one of them being simple, then $\sup_M\mathrm{Scal}\geq 0$ and equality holds if and only if $M$ is a right circular cylinder $\mathbb{R}^{n-1}\times\mathbb{S}^1(r)$, with $r>0$.

Our results in this talk are part of our joint work with S. Carolina García-Martínez, from Universidad Nacional de Colombia in Bogotá, (Geom. Dedicata 156:31--47, 2012) and with Josué Meléndez, from Universidad Autónoma Metropolitana-Iztapalapa in Ciudad de México (Geom. Dedicata 182:117--131, 2016; Geom. Dedicata 199:273--280, 2019).

A Brinkmann spacetime is a Lorentzian manifold $\bar{M}$ that admits a globally defined null vector field $K\in\mathfrak{X}(\bar{M})$ which is parallel. This family of spacetimes contains exact solutions to the Einstein’s field equation that model radiation (electromagnetic or gravitational) moving at the speed of light. Lately, Brinkmann spacetimes have attracted a great deal of attention due to the experimental detection of gravitational waves.

In this work we consider a relevant subfamily of these spacetimes, the plane-fronted waves (in sort, pf-waves). Concretely, we study codimension two spacelike submanifolds, obtaining some interesting results for the case when they are immersed in a pf-wave through a distinguished null hypersurface. As a new approach of these type of results, we also get some conditions which imply that a codimension two spacelike submanifold has to be immersed through such a null hypersurface.

This is part of a work in progress with Alfonso Romero (from University of Granada) and Francisco J. Palomo (from University of Málaga).

Singular Riemannan foliations are generalizations of smooth group isometric group actions. In particular, torus actions by isometries on a given Riemannian manifold have been studied to describe the topology of the manifold, or properties the Riemannian metric might have. We extend this study to the setting of singular Riemannian foliations by tori. We show that some techniques developed for the comparison of two torus actions, can be applied to the foliated setting. For the case when the foliation has codimension 2, we obtain the following result:

Let (M, F) be a singular Riemannian foliation of codimension 2 by tori, on a compact, simply-connected Riemannian manifold. Then the foliation is induced by a smooth torus action.

In the 1980's R. Bartnik proposed a rigidity conjecture on the structure of timelike geodesically complete, globally hyperbolic spacetimes with compact Cauchy hypersurfaces, which if true has impact on our understanding of the general structure of cosmological spacetimes. In this talk I review the progress made in (dis)proving this conjecture by several authors, focusing on some of the most promising recent lines of attack.

In Riemannian geometry, collapse imposes strong geometric and topological restrictions on the spaces on which it occurs. In the case of Alexandrov spaces, which are metric generalizations of complete Riemannian manifolds with a uniform lower sectional curvature bound, collapse is fairly well understood in dimension three. In this talk, I will discuss the geometry and topology of sufficiently collapsed three-dimensional Alexandrov spaces: when the space is irreducible, it is modeled on one of the eight three-dimensional dimensional Thurston geometries, excluding the hyperbolic one. This extends a result of Shioya and Yamaguchi, originally formulated for Riemannian manifolds, to the Alexandrov setting. (Joint work with Luis Guijarro and Jesús Núñez-Zimbrón).

In this talk I will present some constructions of soliton theory with a geometrical flavour. The talk will centre in the space of vortex-antivortex pairs, obtained as stable solutions of the BPS Lagrangian. The BPS Lagrangian appears as a simplification of Skyrme's model from atomic physics by constraining the domain of the fields to the euclidean plane. It was later extended by Sibner et al. to compact Riemann surfaces. There are two types of fundamental solutions of the resulting variational problem, called the vortex and antivortex solution respectively.

The main result is that the moduli space of solutions of Bogomolny equations modulo gauge equivalence is a Riemannian space, in fact, a Kahler manifold, whose metric is incomplete and of finite volume if ambient space is a closed surface. An analytic proof of this statement exists for vortex-antivortex pairs on the sphere and I will talk about the details.

The formulation of the Einstein's field equations as a hyperbolic set of equations leads in a natural way to the problem of the construction of initial data or Cauchy data for the equations. This problem is a highly non-trivial one that indeed gives rise to an independent field of research. The simplest situation occurs in the formulation of the initial data for the vacuum equations where one needs to analyze the vacuum constraint equations. This is a system of differential equations involving the first and the second fundamental form of the initial data hypersurface.

In this work we study how the previous formulation changes when one imposes that the solution of the Cauchy problem is an exact solution known beforehand. This can be formulated in mathematical terms as an existence problem of an isometric embedding of a 3-dimensional Riemannian manifold into a given 4-dimensional Lorentzian manifold. We present explicit results for the case in which the 4-dimensional Lorentzian manifold is a vacuum type D solution and discuss very important particular cases such as the Kerr and the Scwarzschild solutions.

Finally we apply the techniques used for the vacuum Einstein equations to the case of the conformal equations developed by H. Friedrich and construct data whose solution is conformal to a vacuum solution of the Einstein equations. The implications for the analysis of conformal boundaries are discussed.

We show that in any compact Riemannian manifold M, any point is critical for the distance function from some other point q. This extends results of of Bárány, Itoh, Vîlcu and Zamfirescu, who proved a similar statement for Alexandrov surfaces. This is joint work with Fernando Galaz-Garcia.

A Riemannian manifold (M,g) is said to be conformally Einstein if there is an Einstein representative of the conformal class [g]. It was shown by Brinkmann [1] that the existence of a conformally Einstein structure is equivalent to the existence of solution of a (generically overdetermined) PDE. Despite its apparent simplicity, the integration of such equation is surprisingly difficult. This is due, in part, to the fact that the equation is trace-free and divergence-free.

While any two-dimensional manifold is conformally Einstein and three-dimensional manifolds are conformally Einstein if and only if they are conformally flat, there is still a lack of results in the four-dimensional case. There are partial answers in the Kähler situation [2] The classification of conformally Einstein product manifolds has been recently addressed [5].

The purpose of this lecture is to give a complete description of conformally Einstein homogeneous manifolds [3], which provides a natural generalization of previous work of Jensen [4].

[1] H. W. Brinkmann, Riemann spaces conformal to Einstein spaces, Math. Ann. 91 (1924), no. 3-4, 269- -278.

[2] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no.~3, 405--433.

[3] E. Calviño Louzao, X. García-Martínez, E. García-Río, I. Gutiérrez-Rodríguez, and R. Vázquez-Lorenzo, Conformally Einstein and Bach-flat four dimensional homogeneous manifolds, to appear in Journal de Math\'ematiques Pures et Appliqu\'ees (2019).

[4] G. R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geometry 3 (1969), 309--349.

[5] W. Kühnel and H. B. Rademacher, Conformally Einstein product spaces, Differential Geom. Appl. 49 (2016), 65--96.

How can we tell if a spacetime looks more or less like internal Schwarzschild, if there are no Killing fields around--and what ought that to mean?

This is part of a general program aiming at examining a spacetime M purely by means of a foliation F by timelike observers, with no assumptions of symmetry; geometry will be found in terms of time-dependent fields on the space Q of observers (i.e., Q = M/F). In this first stage, the assumed additional structure is that observers are all future-incomplete. The two questions to consider are: (1) What constraints on the time-dependent geometry on Q guarantee that the causal boundary of M is (a) diffeomorphic to Q, (b) spacelike, and (c) attached in the simplest way to M? And (2) what constraints on the geometry of M (such as integral curvature conditions) guarantee or forbid that the spacetime metric has a continuous extension to the causal boundary?

This is somewhat a work still in progress, in that what constitute optimal constraints for (1) is not yet clear, and integral conditions for (2) still being in examination. But the basic outline seems to be one of uniformity constraints on the time-dependent geometry on Q for obtaining the proper causal boundary; and conditions on integration of sectional curvature along observer worldlines for continuous extension of the metric to the boundary.

Our aim in this talk will be to present a new notion for black holes suitable for any strongly causal spacetimes. For this, we will begin by reviewing the classical notion for black holes, which relies on the existence of a conformal boundary. Then, we will show how this classical notion can be extended for any strongly causal spacetime once we consider the causal one instead the conformal one. Finally, we will show that most of the classical properties for black holes remain true, including the fact that closed trapped surfaces are covered by black holes under mild hypotheses.

We review different notions of black holes, their relations to causal and conformal boundaries and some new developments in the field.

The study of lightlike hypersurfaces in a Lorentzian manifold has been a topic of growing interest in the development of Mathematics and Physics of Gravitation. The key difference between the lightlike hypersurfaces and the cases of spacelike or timelike submanifolds arises due to the fact that a lightlike manifold inherits a degenerate symmetric tensor from the ambient metric. Therefore, there is no preferred linear connection on lightlike hypersurfaces.

On the other hand, the notion of Cartan Geometry relates to every homogeneous space $G/H$ (the model) a differential geometric structure.

In this talk, we provide a quick introduction to Cartan Geometries and then, we show how this general frame can be used to a new approach to the study of the lightlike manifolds.

In this talk we present the notion of a screen quasi-conformal null hypersurface and provide examples of its use in classifying isoparametric and Einstein null hypersurfaces in GRW-spacetimes. This is joint work with M. Navarro (UADY) and O. Palmas (UNAM).

We deal with minimal surfaces in a sphere which are locally isometric to a pseudoholomorphic surface of $\mathbb{S}^{5}$ in the nearly Kaehler $\mathbb{S}^6$. Besides flat minimal surfaces in spheres, direct sums of members of the associate family of pseudoholomorphic surface in $\mathbb{S}^5$ are locally isometric to a pseudoholomorphic surface in $\mathbb{S}^5$. In order to classify all such surfaces we investigate certain invariants of geometric significance, the Hopf differentials, and their holomorphicity. Using higher fundamental forms and higher curvature ellipses we are led to the study a class of minimal surfaces, which are called exceptional. We show that exceptional minimal surfaces are related to our problem regarding a Ricci-like condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which are locally isometric to a pseudoholomorphic surface of $\mathbb{S}^{5}$, are exceptional. Thus, the classification of these minimal surfaces is reduced to the classification of exceptional minimal surfaces that are locally isometric to a pseudoholomorphic surface in $\mathbb{S}^5$. In fact, we provide an affirmative answer for exceptional minimal surfaces in odd dimensional spheres.