Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics

We establish new existence results for the Einstein constraint equations for mean extrinsic curvature arbitrarily far from constant. The results hold for rescaled background metric in the positive Yamabe class, with freely specifiable parts of the data sufficiently small, and with matter energy density not identically zero. Two technical advances make these results possible: A new topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new global supersolution construction for the Hamiltonian constraint that is similarly free of such conditions. The results are presented for strong solutions on closed manifolds, but also hold for weak solutions and for compact manifolds with boundary. These results are apparently the first that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.