  Given asymptotically flat initial data on M^3 for the vacuum Einstein field
equation, and given a bounded domain in M, we construct solutions of the vacuum
constraint equations which agree with the original data inside the given
domain, and are identical to that of a suitable Kerr slice (or identical to a
member of some other admissible family of solutions) outside a large ball in a
given end. The data for which this construction works is shown to be dense in
an appropriate topology on the space of asymptotically flat solutions of the
vacuum constraints. This construction generalizes work of the first author,
where the time-symmetric case was studied.
