CONSTANT MEAN-CURVATURE SOLUTIONS OF THE EINSTEIN CONSTRAINT EQUATIONS ON CLOSED MANIFOLDS

We prove in detail a theorem which completes the evaluation and parame trization of the space of constant mean curvature (CMC) solutions of t he Einstein constraint equations on a closed manifold. This theorem de termines which sets of CMC conformal data allow the constraint equatio ns to be solved, and which sets of such data do not. The tools we desc ribe and use here to prove these results have also been found to be us eful for the study of non-constant mean curvature solutions of the Ein stein constraints.