Proof of the local mass-angular momenta inequality for $U{(1)}^{2}$ invariant black holes

We consider initial data for extreme vacuum asymptotically flat black holes with symmetry. Such geometries are critical points of a mass functional defined for a wide class of asymptotically flat, ‘’ symmetric maximal initial data for the vacuum Einstein equations. We prove that the above extreme geometries are local minima of mass among nearby initial data (with the same interval structure) with fixed angular momenta. Thus the ADM mass of …