Mathematical general relativity merges together active areas of research in partial differential equations, differential geometry and geometric analysis. One century after its appearance, a number of fundamental mathematical questions concerning general relativity remain unanswered. Their future resolution will be of great interest both in developing new mathematical tools, at the crossroads of Mathematical physics and geometric analysis, and in understanding the applicability and limitations of the theory.
This workshop aims at providing an opportunity for researchers to gather in Lyon to report on the most recent advances in the field.
Wheeler’s thin-sandwich conjecture proposes a particular way of parametrising the space of solutions of the Einstein constraint equations, which is deeply motivated by physical considerations. The reduced thin-sandwich equations appear in this context as a method of solving the Einstein constraint equations in terms of the lapse function and shift vector field. Some influential papers by Wheeler and his collaborators conjectured that this method was general enough so as to parametrise generic solutions of the constraint equations. During this talk, we will discuss some recent results and show that on any compact n-dimensional manifold, n ≥ 3, there is an open subset in the space of solutions of the constraint equations where the thin-sandwich problem is well-posed. Furthermore, we will discuss natural geometric conditions that lead to existence of solutions and analyse the extent of generality of such conditions. Finally, we will analyse how this problem translates to asymptotically euclidean manifolds.
Le théorème d'énergie positive hyperbolique affirme que toute variété riemannienne complète, asymptotique à l'espace hyperbolique réel, et dont la courbure scalaire est minorée par celle du modèle, possède un vecteur énergie-impulsion de genre temps dirigé vers le futur, ce vecteur étant nul seulement pour le modèle. Nous verrons une preuve de ce résultat en toutes dimensions et sans condition spin. Il s'agit d'un travail en collaboration avec Piotr Chrusciel.
We will begin with a review of the classical Oppenheimer-Snyder model (1939) on gravitational collapse in spherical symmetry. We will discuss why this model is in itself incomplete and pass on to Christodoulou’s two-phase model (1995). The main part of the talk is concerned with one possible end state of the latter model: hard stars. These are idealized models of neutron stars. Variational properties and linearised dynamics will be discussed, in particular their relevance to the orbital stability problem in spherical symmetry. Various obstacles to a global in time result are outlined, in particular the absence of a dispersion mechanism, the trapped surface formation scenario due to reflecting boundary conditions (cf. AdS-scalar field) and the possibility of phase transitions within the two phase model to avoid Rayleigh-Taylor instabilities. This is a joint work with Volker Schlue at the University of Melbourne.
The mass of an asymptotically hyperbolic manifold is a vector in Minkowski space defined in terms of the geometry at infinity of the manifold. It enjoys covariance properties under the change of coordinate chart at infinity. In this talk we classify covariants satisfying similar properties. This is a join work with J. Cortier and M. Dahl.
The product spacetime $AdS_2 \times S^2$ arises, e.g., as the near horizon geometry of the extremal Reissner-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence. In my talk I will present joint work with Greg Galloway studying rigidity properties of asymptotically $AdS_2 \times S^2$ spacetimes satisfying the null energy condition: Any such spacetime must contain two continuous transversal foliations by totally geodesic null hypersurfaces intersecting in isometric, totally geodesic round 2-spheres. However, concrete examples show that this is not enough to force the spacetime to be isometric to $AdS_2 \times S^2$ and it is an open problem what minimal additional assumption could be made to guarantee this. I will also briefly touch upon current work in progress toward a conformal definition of ''asymptotically $AdS_2 \times S^2$ ends'' using the notion of a ’singular scri’.
Inspired by the Mantoulidis and Schoen construction, we obtain time-symmetric black hole initial data sets for the Einstein--Maxwell equations satisfying the dominant energy condition, such that their horizon boundary geometry is prescribed, and their total masses and total charges are controlled. We also formulate an ad-hoc notion of boundary Bartnik mass in this context and compute its value for minimal Bartnik data. This talk is based on a joint work with A. Alaee and C. Cederbaum.
In this mini-course we will discuss recent topics in the analysis of the conformal constraint system of equations in the focusing case, that is with matter or in the presence of a positive cosmological constant. We will describe the conformal method and recall the available results for the conformal constraint system. Emphasis will then be put on compactness and stability properties of the system, and in particular on their very recent applications to new existence results.
While the concepts of mass and linear momentum are by now well-established in mathematical general relativity, it is not entirely clear whether the existing notions of center of mass are the ultimate ones. In this talk we introduce a novel geometric foliation of an initial data set and discuss its connections to and advantages over the existing approaches to defining the center of mass of an isolated system. This is joint work with Carla Cederbaum.
We study initial data in General Relativity, which are defined as solutions to the constraint equations. The focus in this talk is a modified version of the conformal method proposed by David Maxwell. While the model seems more strongly justified from a geometrical standpoint, the resulting system becomes significantly more difficult to solve; it presents critical nonlinear terms, including gradient terms. We work in dimensions 3,4 and 5, under smallness assumptions and in the presence of a scalar field with positive potential. The tools we use are related to obtaining a priori estimates (compactness results) and a fixed-point theorem.
A conformal pseudo-Finsler structure on a manifold consists essentially in giving a field of tangent cones: a convex proper cone in each tangent space. We are interested in the automorphism group of such a structure, particularly in the case where this group has a strong dynamics.