  We give a sufficient condition, with no restrictions on the mean curvature,
under which the conformal method can be used to generate solutions of the
vacuum Einstein constraint equations on compact manifolds. The condition
requires a so-called global supersolution but does not require a global
subsolution. As a consequence, we construct a class of solutions of the vacuum
Einstein constraint equations with freely specified mean curvature, extending a
recent result of Holst, Nagy, and Tsogtgerel [HNT07] which constructed similar
solutions in the presence of matter. We give a second proof of this result
showing that vacuum solutions can be obtained as a limit of [HNT07] non-vacuum
solutions. Our principal existence theorem is of independent interest in the
near-CMC case, where it simplifies previously known hypotheses required for
existence.
