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We construct transformations which take asymptotically AdS hyperbolic initialdata into asymptotically flat initial data, and which preserve relevantphysical quantities. This is used to derive geometric inequalities in theasymptotically AdS hyperbolic setting from counterparts in the asymptoticallyflat realm, whenever a geometrically motivated system of elliptic equationsadmits a solution. The inequalities treated here relate mass, angular momentum,charge, and horizon area.

Transformations of Asymptotically AdS Hyperbolic Initial Data and Associated Geometric Inequalities