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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)**

**Lecture room 15-25 326 (third floor)**

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

**Abstract. W**e 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.