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

Abstract: I prove a convergence theorem in the context of $3+1$ dimensional vacuum Einstein's equations including a positive cosmological constant on spacetimes $\widetilde{M}$ of topological type $M\times \mathbb{R}$, where $M$ is a closed connected oriented Riemannian $3-$manifold of negative Yamabe type. The theorem concerns the global existence of the solutions of the vacuum Einstein$-\Lambda$ equations for arbitrary large initial data and convergence of the spatial geometry to a constant negative scalar curvature metric in a constant mean extrinsic curvature (CMC) transported spatial gauge. This theorem could be interpreted as a weak cosmic censorship result in the cosmological framework under \textit{generic} initial data. An important corollary of the theorem is the failure of cosmological principle as a derived consequence of Einstein's equations including a positive cosmological constant as well as the failure of geometrization of spatial slice in the sense of Thurston. A similar theorem is stated for positive Yamabe slices under a technical assumption.