On the Area of the Symmetry Orbits of the Einstein-Vlasov-Scalar Field System with Plane and Hyperbolic Symmetry

prove in the case of cosmological models for the Einstein-Vlasovscalar field system, that the area radius of compact hypersurfaces tends to a constant value as the past boundary of the maximal Cauchy development is approached. In other case, there is at least one Cauchy hypersurface of constant areal time coordinate in plane and hyperbolic symmetric spacetimes. Moreover, we show that the areal time coordinate R = t which covers these spacetimes runs from zero at infinity with the singularity occuring at R = 0. The sources of the equations are generated by a distribution function and a massless scalar field, subject to the Vlasov and wave equations respectively.