Cosmic Censorship near FLRW spacetimes with negative spatial curvature

We consider general initial data for the Einstein scalar-field system on a closed $3$-manifold $(M,γ)$ which is close to data for a Friedman-Lemaître-Robertson-Walker solution with homogeneous scalar field matter and a negative Einstein metric $γ$ as spatial geometry. We prove that the maximal globally hyperbolic development of such initial data in the Einstein scalar-field system is past incomplete in the contracting direction and exhibits stable collapse into a Big Bang curvature singularity. Under an additional condition on the first positive eigenvalue of $-Δ_γ$ satisfied, for example, by closed hyperbolic 3-manifolds of small diameter, we prove that the data evolves to a future complete spacetime in the expanding direction which asymptotes to a vacuum Friedman solution with $(M,γ)$ as the expansion normalized spatial geometry. In particular, the Strong Cosmic Censorship conjecture holds for this class of solutions in the $C^{2}$-sense.