#### 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
##### 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 $\Lambda >0$, 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 $\Lambda \ge 0$. In all cases, there is no restriction on the size of initial data.

##### Keywords
Einstein equations, singularities, hyperbolic partial differential equations
Primary: 83C05