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, ) which is close to data for a Friedman-Lematre-Robertson-Walker solution with homogeneous scalar field matter and a negative Einstein metric 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 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, ) as the expansion normalized spatial geometry.In particular, the strong cosmic censorship conjecture holds for this class of solutions in the C 2 -sense.1. Introduction 2. Big bang stability: preliminaries 3. Big bang stability: norms, energies and bootstrap assumptions 4. Big bang stability: a priori estimates 5. Big bang stability: elliptic lapse estimates 6.Big bang stability: energy and norm estimates 7. Big bang stability: improving the bootstrap assumptions 8. Big bang stability: the main theorem 9.