Structure of solutions near the initial singularity for the surface-symmetric Einstein-Vlasov system

Results on the behaviour in the past time direction of cosmological models with collisionless matter and a cosmological constant $\mathrm{\Lambda }$ are presented. It is shown that under the assumption of non-positive $\mathrm{\Lambda }$ and spherical or plane symmetry the area radius goes to zero at the initial singularity. Under a smallness assumption on the initial data, these properties hold in the case of hyperbolic symmetry and negative $\mathrm{\Lambda }$ as well as in the positive $\mathrm{\Lambda }$ case. Furthermore in the latter cases past global existence of spatially homogeneous solutions is proved for generic initial data. The early-time asymptotics is shown to be Kasner-like for small data.