  Generalising an analysis of Corvino and Schoen, we study surjectivity
properties of the constraint map in general relativity in a large class of
weighted Sobolev spaces. As a corollary we prove several perturbation, gluing,
and extension results: we show existence of non-trivial, singularity-free,
vacuum space-times which are stationary in a neighborhood of $i^0$; for small
perturbations of parity-covariant initial data sufficiently close to those for
Minkowski space-time this leads to space-times with a smooth global Scri; we
prove existence of initial data for many black holes which are exactly Kerr --
or exactly Schwarzschild -- both near infinity and near each of the connected
components of the apparent horizon; under appropriate conditions we obtain
existence of vacuum extensions of vacuum initial data across compact
boundaries; we show that for generic metrics the deformations in the
Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on
the connected sum manifold coincide with the original ones except for a small
neighborhood of the gluing region; we prove existence of asymptotically flat
solutions which are static or stationary up to $r^{-m}$ terms, for any fixed
$m$, and with multipole moments freely prescribable within certain ranges.
