arXiv2024
In this paper, we study the past asymptotics of $(2+1)$-dimensional solutions to the Einstein scalar-field Vlasov system which are close to Friedman-Lema\^itre-Robertson-Walker spacetimes on an initial hypersurface diffeomorphic to a closed orientable surface $M$ of arbitrary genus. We prove that such solutions are past causally geodesically incomplete, form a curvature singularity and exhibit stable Kretschmann scalar blow-up in the contracting direction. In particular, the spacetime is $C^2$-inextendible towards the past where causal geodesics become incomplete. Moreover, we show that geometry and matter are asymptotically velocity term dominated toward the past, remaining close to their background counterparts. When viewed on the co-mass shell, the Vlasov distribution in particular converges to a limiting distribution on the Big Bang hypersurface, while asymptotics of the spatial metric lead to slight degeneracies when trying to control the components of the Vlasov energy-momentum tensor. These also manifest when viewing the distribution function on the mass shell, where velocities generally approach a smooth one-dimensional subbundle of the tangent bundle, leading the distribution to become highly anisotropic, when viewed in terms of the geometry induced by the constant curvature spatial reference metric. Compared to previous results in higher dimensions, in particular [FU25b], inhomogeneous terms in the wave and Vlasov equations factor in more strongly in our setting, which a priori creates additional hurdles at high orders while largely keeping the quiescent system intact. As a corollary, our main result shows that the Strong Cosmic Censorship conjecture holds for certain polarized $U(1)$-symmetric solutions to the Einstein vacuum equations that emanate from a spatial hypersurface diffeomorphic to $M\times\mathbb{S}^1$.