Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature

We show that asymptotically hyperbolic initial data satisfying smallness conditions in dimensions $nge 3$, or fast decay conditions in $nge 5$, or a genericity condition in $nge 9$, can be deformed, by a deformation which is supported arbitrarily far in the asymptotic region, to ones which are exactly Kottler (Schwarzschild- adS) in the asymptotic region.