Solution of the Einstein constraints on non-maximal submanifolds

It has long been known that, by use of the 'conformal method', the Einstein constraints on an initial manifold split into a linear elliptic system and a nonlinear elliptic (Lichnerowicz) equation in the case where the manifold has constant mean extrinsic curvature. However there are spacetimes which do not admit such submanifolds (cf. Brill, Bartnik). In a recent paper, in collaboration with J. Isenberg and V. Moncrief, the author proved existence theorems for the coupled system of constraints, on an initial compact manifold with non-constant mean extrinsic curvature. Here the author considers the non-compact, asymptotically Euclidean case of non-maximal initial submanifolds.