We prove a semi-global existence and convergence theorem for the $(3+1)$-dimensional vacuum Einstein equations with positive cosmological constant on spacetimes $\widetilde{ M} \sim M \times \mathbb{ R}$, 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 a class of arbitrarily large initial data converge to a Riemannian metric of constant negative scalar curvature in infinite Newtonian-like `time'. A main novelty is to uncover a new weak null-type structure (different from the well known null structure in the literature) in the field equations induced by the positive cosmological constant in constant mean curvature gauge that is absent in pure vacuum. As a consequence, the Einstein-$\Lambda$ flow generically fails to produce geometrization in the sense of Thurston. Our results affirm a conjecture of Ringstr\"om concerning the asymptotic in-distinguishability of spatial topology in the large data regime of Einstein-$\Lambda$ dynamics. A related result is established for positive Yamabe type under a technical condition.
Categories: gr-qc math-ph math.AP math.DG math.MP