Seminar on

Mathematical General Relativity


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

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

ANR Project

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

Thursday March 22, 2012

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

Lecture room 15-25 102 (first level)

14h   Mihalis Dafermos (Cambridge) Black holes without spacelike singularities

Abstract.   It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordstroom data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably, in fact, cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner-Nordstrom, there is no spacelike component of either the future or the past singularity.

15h30  Rabah Souam (Paris)  Harmonic diffeomorphisms and maximal surfaces

Abstract.  We study the existence (or the non-existence) of harmonic diffeomorphisms between certain domains in the Euclidean  two-sphere. In particular, we construct harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs with isolated singularities in the Lorentzian product M x R, where M is an arbitrary n-dimensional compact Riemannian manifold (with n larger than 1).  In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the (once) punctured sphere, and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean two-sphere. This is a joint work with Antonio Alarcon.