Vol. 4, No. 2, 2011

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On the area of the symmetry orbits of cosmological spacetimes with toroidal or hyperbolic symmetry

Jacques Smulevici

Vol. 4 (2011), No. 2, 191–245

We prove several global existence theorems for spacetimes with toroidal or hyperbolic symmetry with respect to a geometrically defined time. More specifically, we prove that generically, the maximal Cauchy development of T2-symmetric initial data with positive cosmological constant Λ > 0, in the vacuum or with Vlasov matter, may be covered by a global areal foliation with the area of the symmetry orbits tending to zero in the contracting direction. We then prove the same result for surface symmetric spacetimes in the hyperbolic case with Vlasov matter and Λ 0. In all cases, there is no restriction on the size of initial data.

Einstein equations, singularities, hyperbolic partial differential equations
Mathematical Subject Classification 2000
Primary: 83C05
Received: 1 September 2009
Revised: 24 December 2009
Accepted: 31 July 2010
Published: 18 November 2011
Jacques Smulevici
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 1
D-14476 Potsdam-Golm
Laboratoire de Mathématiques
Université Paris-Sud 11, bât. 425
91405 Orsay