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Abstract

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 T 2-symmetric initial data with positive cosmological constant > 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 ≥ 0. In all cases, there is no restriction on the size of initial data. 199 6. Remarks on the strategy of the proofs 200 7. Proof of Theorem 1 204 8. Proof of Theorem 2 216 9. Proof of Theorem 3 235 10. Comments and open questions 240 Appendix A. Initial data and the constraint equations 242 Appendix B. Surface-symmetric spacetimes in areal coordinates 243 Appendix C. From symmetric initial data to symmetric spacetimes 243 Acknowledgements 244 References 244 1 See [Choquet-Bruhat 1970; 1971] for the case of Vlasov matter. 2 The conjecture was originally developed by R. Penrose [1979] and first formulated as a statement about the global geometry of the maximal Cauchy development in [Moncrief and Eardley 1981]. See also the presentation in [Christodoulou 1999]. 3 The regularity concerns here the degree of differentiability of the possible extensions and gives rise to different versions of the conjecture. For instance, the C 2 formulation of the conjecture is obtained by replacing "regular" with C 2 in the above statement of the conjecture. 4 They also contain the even more special case of polarized T 3-Gowdy spacetimes. 5 Note that the plane symmetric case is a special case of T 3-Gowdy polarized solutions. 9 Compared to the usual metric for these spacetimes, we use the square of the radius function t = r 2 as the time coordinate rather than the radius function r itself. Moreover, we have introduced the functions α and ν by analogy with the T 2 case, so as to ease the application of the method of the T 2 case to this class of spacetimes. See Appendix B for a description of the change of coordinates from the usual parametrization to this one. SYMMETRY ORBITS OF COSMOLOGICAL SPACETIMES WITH TOROIDAL OR HYPERBOLIC SYMMETRY 197 2C. The Einstein-Vlasov system. Apart from the vacuum case, where we will set T µν = 0 in (1), we will couple the Einstein equations to the Vlasov matter model, which we present in this section. Let ᏼ ⊂-ᏹ denote the set of all future-directed timelike vectors of length −1. ᏼ is classically called the mass shell. Let f denote a nonnegative function on the mass shell. The Vlasov equation for f is derived from the condition that f be preserved along geodesics. In coordinates, we therefore have p α ∂ x α f − α βγ p β p γ ∂ p α f = 0, 200 JACQUES SMULEVICI Theorem 3. Let (ᏹ, g, f) be the maximal development of k = −1 surface-symmetric initial data with Vlasov matter and ≥ 0. Suppose that the Vlasov field f does not vanish identically. Then (ᏹ, g) adm...