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We consider several geometric inequalities in general relativity involvingmass, area, charge, and angular momentum for asymptotically hyperboloidalinitial data. We show how to reduce each one to the known maximal (or timesymmetric) case in the asymptotically flat setting, whenever a geometricallymotivated system of elliptic equations admits a solution.

Reduction Arguments for Geometric Inequalities Associated With Asymptotically Hyperboloidal Slices