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Seminar on

Mathematical General Relativity

Organizers: Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Univ. Pierre et Marie Curie) Gabriele Veneziano (Collège de France)

 

With the financial support of  the Fondation des Sciences Mathématiques de Paris


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Monday March 7, 2011

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

Lecture rooms: 15/25  1-03 (morning) and 1-02 (afternoon)

 

11h   Jacques Smulevici (AEI, Potsdam)  Weakly regular T2 symmetric spacetimes

Abstract. Under weak regularity assumptions, only, we develop a fully geometric theory of vacuum Einstein spacetimes with T2–symmetry, establish the global well-posedness of the initial value problem for Einstein’s field equations, and investigate the global causal structure of the constructed spacetimes. Our weak regularity assumptions are the minimal ones allowing to give a meaning to the Einstein equations under the assumed symmetry and to solve the initial value problem. First of all, we introduce a frame adapted to the symmetry in which each Christoffel symbol can be checked to belong to Lp for some p. We identify certain cancellation properties taking place in the expression of the Riemann and Ricci curvatures, and this leads us to a reformulation of the initial value problem for the Einstein field equations when the initial data set has weak regularity. Second, we investigate the future development of a weakly regular initial data set. We check that the area R of the orbits of symmetry must grow to infinity in the future timelike directions, and we establish the existence of a global foliation by the level sets of the function R. Our weak regularity assumptions only require that R is Lipschitz continuous while the metric coefficients describing the initial geometry of the orbits of symmetry are in the Sobolev space H1 and the remaining coefficients have even weaker regularity. We develop here the compactness arguments required to cover the natural level of regularity associated with the energy of the system of partial differential equations determined from Einstein’s field equations. This is a joint work in collaboration with P.G. LeFloch (Paris).

14h   Erwann Delay (Univ. Avignon) Recollement des TT tenseurs

Abstract. La méthode de Corvino-Schoen permet de recoller localement et de façon lisse deux solutions des équations de contraintes. Nous verrons que l’on peut d’une façon générale recoller deux éléments du noyau de certains opérateurs à symbole surjectif et non injectif. Par exemple on peut recoller deux champs de vecteurs à divergence nulle ou deux TT-tenseurs. On en déduit par exemple que sur toute boule riemannienne ouverte, l’ensemble des TT-tenseurs lisses à support compact est de dimension infinie. On simplifie ainsi la construction de données initiales CMC.