Constant mean curvature solutions of the Einstein constraint equations on closed manifolds

We prove in detail a theorem which completes the evaluation and parametrization of the space of constant mean curvature (CMC) solutions of the Einstein constraint equations on a closed manifold. This theorem determines which sets of CMC conformal data allow the constraint equations to be solved, and which sets of such data do not. The tools we describe and use here to prove these results have also been found to be useful for the study of non-constant mean curvature solutions of the Einstein constraints.