You are currently browsing the category archive for the ‘GENERAL RELATIVITY’ category.
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Organizers
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday March 4, 2015
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25–326
14h François Fillastre (Cergy-Pontoise) Minkowski problem in Minkowski space
Abstract. T. Barbot, F. Beguin and A. Zeghib solved a smooth Lorentzian version of the Minkowski problem in dimension (2+1). More precisely they proved that if M is a flat 3-dimensional maximal globally hyperbolic spatially compact spacetime, then there exists a unique strictly convex smooth space-like surface in M with a prescribed smooth positive Gauss curvature. We will look at this problem for any dimensions. The existence part is solved in a generalized way (a measure is prescribed rather than a function). Concerning the regularity of the solution, the 2+1 case is specific. The arguments are based on tools from the geometry of convex sets. Joint work with Francesco Bonsante (Pisa).
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Organizers
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday Feb. 11, 2015
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25–326
14h Bruno Premoselli (Cergy-Pontoise) Robustness of the conformal constraints in a scalar-field setting
Abstract. The constraint equations arise in the initial-value formulation of the Einstein equations. The conformal method allows one to rewrite the constraint equations into a determined system of nonlinear, supercritical, elliptic PDE’s. In this talk, we will investigate some stability properties for this elliptic system. The notion of stability under consideration, defined as the continuous dependence of the set of solutions of the conformal constraint system with respect to its coefficients, is reformulated for the conformal method. The analysis of these stability properties involves blow-up techniques concerning defects of compactness for supercritical nonlinear elliptic equations. This is a joint work with Olivier Druet.
15h30 Christophe Bavard (Bordeaux) Points conjugués des tores lorentziens
Abstract. Les points conjugués jouent un rôle important dans l’étude des variétés riemanniennes et lorentziennes, en particulier pour l’étude du rayon d’injectivité. Dans le cadre riemannien, l’absence de points conjugués impose des contraintes assez fortes sur la topologie de la variété, et parfois même sur sa géométrie. Ainsi, un résultat de Hopf (1948), généralisé par Burago et Ivanov (1994), affirme qu’un tore riemannien sans points conjugués est nécessairement plat. Dans cet exposé, je montrerai l’existence de métriques sans points conjugués dans toute composante connexe de l’espace des métriques lorentziennes sur le tore de dimension 2 ; cela prouve en particulier l’existence de tores lorentziens sans points conjugués et non plats. Il s’agit d’un travail conjoint avec Pierre Mounoud.
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Organizers
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday Nov. 26, 2014
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25–104
14h Qian Wang (Oxford) A geometric approach to the sharp local well-posedness theory for quasilinear wave equations
Abstract. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces Hs with s>2+(2-\sqrt 3)/2 for general quasilinear wave equation in (1+3) spacetime by him and Rodnianski. Via this approach they obtained the local well-posedness with s>2 for (1+3) vacuum Einstein equations. A proof of the sharp H2+ local well-posedness result for general quasilinear wave equation was provided by Smith and Tataru by constructing a parametrix using wave packets. The difficulty of the problem is that one has to face the major hurdle caused by the Ricci tensor of the metric for the quasilinear wave equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in (1+3) spacetime by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.
15h30 Jonathan Luk (Cambridge, UK) Stability of the Kerr Cauchy horizon
Abstract. The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.
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Organizers
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday Sept. 17, 2014
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 15-25-321
14h Arick Shao (Imperial College) Unique continuation from infinity for linear waves
Abstract. We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects. This is joint work with Spyros Alexakis and Volker Schlue.
15h30 Claude Warnick (Warwick) Dynamics in anti-de Sitter spacetimes
Abstract. When solving Einstein’s equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike hypersurface. Questions of local well-posedness and global stability are sensitive to the choices of boundary conditions. I will present recent work exploring the effects of non-trivial boundary data for the asymptotically AdS initial-boundary value problem, including a recent result in collaboration with Holzegel. I will also outline some interesting open problems in the area.
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Seminar on
Mathematical General Relativity
Organizers:
Philippe G. LeFloch (Paris)
Jérémie Szeftel (Paris)
Ghani Zeghib (Lyon)
ANR Project
“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”
Wednesday June 25, 2014
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Lecture room 1525-321
14h Jan Sbierski (Cambridge, UK) A Zorn-free proof of the existence of a maximal Cauchy development for the Einstein equations
Abstract. In 1969, Choquet-Bruhat and Geroch showed that there exists a unique maximal Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In particular, their proof ensures the existence of the maximal development without actually constructing it. In this talk, we present a proof of the existence of a maximal Cauchy development that avoids the use of Zorn’s lemma and, moreover, provides an explicit construction of the maximal development.
15h15 Sergiu Klainerman (Princeton) Remarks on the stability of the Kerr solution in axial symmetry
Mathematical general relativity, compressible fluids, and more 