A large data result for vacuum Einstein's equations

We prove a global well-posedness and asymptotic convergence theorem for the $(3+1)$-dimensional vacuum Einstein equations with positive cosmological constant $\Lambda$ on globally hyperbolic spacetimes $\widetilde M \cong M \times \mathbb R$, where $M$ is a closed three-manifold of negative Yamabe type. In constant-mean-curvature transported spatial coordinates, an open set of large initial data gives rise to future-global solutions whose renormalized spatial metrics converge smoothly to a limiting metric of constant negative scalar curvature. The key new ingredient is an integrable damping mechanism, induced by the cosmological constant in this gauge and absent in the $\Lambda=0$ vacuum problem, which yields time-integrable decay for the nonlinear evolution. As a consequence, the Einstein--$\Lambda$ flow does not in general canonically encode the Thurston geometrization of the underlying three-manifold. This confirms a conjecture of Ringstr\"om on the asymptotic topological indistinguishability of large-data Einstein--$\Lambda$ dynamics. An analogous theorem is also proved for manifolds of positive Yamabe type, under an additional technical hypothesis.