Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature

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