A de Sitter no-hair theorem for 3+1d cosmologies with isometry group forming 2-dimensional orbits

We study, using Mean Curvature Flow methods, 3+1 dimensional cosmologies with a positive cosmological constant, matter satisfying the dominant and the strong energy conditions, and with spatial slices that can be foliated by 2-dimensional surfaces that are the closed orbits of a symmetry group. If these surfaces have non-positive Euler characteristic (or in the case of 2-spheres, if the initial 2-spheres are large enough) and also if the initial spatial slice is expanding everywhere, then we prove that asymptotically the spacetime becomes physically indistinguishable from de Sitter space on arbitrarily large regions of spacetime. This holds true notwithstanding the presence of initial arbitrarily-large density fluctuations.