A large data semi-global existence and convergence theorem for vacuum Einstein's equations

We prove a convergence theorem for the $(3+1)$-dimensional vacuum Einstein equations with positive cosmological constant on spacetimes $\widetilde{M} \cong M \times \mathbb{R}$, where $M$ is a closed, connected, oriented three-manifold of negative Yamabe type. In constant mean curvature transported spatial coordinates, we show that solutions arising from arbitrarily large initial data converge to a Riemannian metric of constant negative scalar curvature in infinite Newtonian-like' time'. As a consequence, the Einstein-$Λ$ flow generically fails to produce geometrization in the sense of Thurston and violates the cosmological principle. Our results affirm a conjecture of Ringström concerning the asymptotic indistinguishability of spatial topology in the large data regime. A related result is established for positive Yamabe type under a technical condition.