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The drift method, introduced by the second author, provides a new formulationof the Einstein constraint equations, either in vacuum or with matter fields.The natural of the geometry underlying this method compensates for its slightlygreater analytic complexity over, say, the conformal or conformal thin sandwichmethods. We review this theory here and apply it to the study of solutions ofthe constraint equations with non-constant mean curvature. We show that thismethod reproduces previously known existence results obtained by other methods,and does better in one important regard. Namely, it can be applied even whenthe underlying metric admits conformal Killing (but not true Killing) vectorfields. We also prove that the absence of true Killing fields holdsgenerically.

Conformal Fields and the Structure of the Space of Solutions of the Einstein Constraint Equations