Category Archive

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

With the financial support of the ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday January 5, 2012

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25 326 (third floor)

14h  Alain Bachelot (Bordeaux) Klein-Gordon equation on the Anti-de Sitter universe AdS5

Abstract.  We consider the Klein-Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass is larger than −1, the Cauchy problem is well posed despite the loss of global hyperbolicity due to the time-like horizon. We express all finite energy solutions in the form of a continuous Kaluza-Klein tower. We investigate the case of gravitational fluctuations, and  electromagnetic waves. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish several results on the asymptotic behavior: dispersive estimates, global Strichartz estimates, existence of a lacuna, equi-partition of the energy. We address the cosmological model of the `negative tension’ Minkowski brane. We prove that the hyperbolic mixed problem is well-posed and that all normalizable solutions can be expanded in a discrete Kaluza-Klein tower. Finally, we obtain some L2−L∞ estimates in suitable weighted Sobolev spaces.

15h30 Gilles Carron (Nantes) Rigidity of critical metrics

Abstract. We explain how an elementary idea (existence of bubble of curvature) can be used to proved new and old rigidity results for critical metrics. For instance, we re-prove an old result by M. Anderson that, for an Einstein metric, we get a control on the curvature from a control on the volume.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

With the financial support of the ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday Nov. 24, 2011

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25 326 (third floor)

14h   Paul Laurain (Paris 7)  Surfaces with constant mean curvature in a Riemannian manifold of  dimension 3

Abstract.  The surfaces with constant mean curvature (CMC) in a spacelike hypersurface are geometrically and physically very interesting, as shown by Huisken and Yau in 1996 or in the beautiful thesis of Bray. However, the purpose of this talk is not to develop the physical properties of CMC surfaces but to see on an example what are the analytical difficulties encountered when studying these surfaces. In fact, we will  show how to study CMC surfaces in terms of partial differential equations in order to derive geometric properties. We emphasize in particular the key difficulties generated by the conformal invariance of the problem as the phenomena of concentration and we will show how the structure of the equation helps us to overcome them.

15h30  James D.E. Grant (Vienna)  Null injectivity radius estimates

Abstract. I will report on joint work with P.G. LeFloch, in which we use comparison techniques, such as the Rauch comparison theorem and Hessian comparison theorem, to estimate the null injectivity radius on a Lorentzian manifold. This work gives a more geometrical setting for work of Klainerman and Rodnianski on null injectivity radius estimates.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

With the financial support of the ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Thursday November 3, 2011

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25 326 (third floor)

14h   Charles Boubel (Strasbourg) Germs of Lorentzian metrics and holonomy

Abstract.  The holonomy group of a pseudo-Riemannian metric g -so e.g. a Riemannian or a Lorentzian metric- is a subgroup of O(g) which indicates, in a certain sense, how much its Levi-Civita connection fails to be flat. A central task related to those groupes is to determine the list of the subgroups of O(g) arising as holonomy, and for each item, to parametrize the set of corresponding metrics and build global examples (i.e. complete or compact). In the Riemannian case, this work is now done. We will see that, regarding holonomy matters, Lorentzian metrics behave totally differenly from Riemannian ones. I will review works of L. Bérard-Bergery, A. Ikemakhen, T. Leistner, A. Galaev, and myself, that together deal with the local aspect of the question.

15h30  David Parlongue (Nice) Breakdown criteria and extendibility in general relativity

Abstract. We will begin this talk by reviewing a geometric breakdown criterion for Einstein’s vacuum equations introduced by S. Klainerman and I. Rodnianski and various improvements (non-vacuum case, integral conditions, various gauge choices). We will then examine a spacetime localization of these criteria. We will focus on consequences in terms of formation of singularities, extendibility of spacetimes, and local regularity of foliations.