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

A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature