  We show that asymptotically hyperbolic initial data satisfying smallness
conditions in dimensions $n\ge 3$, or fast decay conditions in $n\ge 5$, or a
genericity condition in $n\ge 9$, 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.
