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When working with asymptotically hyperbolic initial data sets for generalrelativity it is convenient to assume certain simplifying properties. We provethat the subset of initial data sets with such properties is dense in the setof physically reasonable asymptotically hyperbolic initial data sets. Morespecifically, we show that an asymptotically hyperbolic initial data set withnon-negative local energy density can be approximated by an initial data setwith strictly positive local energy density and a simple structure at infinity,while changing the mass arbitrarily little. This is achieved by suitablymodifying the argument used by Eichmair, Huang, Lee and Schoen in theasymptotically Euclidean case.

A density theorem for asymptotically hyperbolic initial data satisfying the dominant energy condition