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We consider initial data for extreme vacuum asymptotically flat black holeswith $\mathbb{R} \times U(1)^2$ symmetry. Such geometries are critical pointsof a mass functional defined for a wide class of asymptotically flat,`$(t-\phi^i)$' symmetric maximal initial data for the vacuum Einsteinequations. We prove that the above extreme geometries are local minima of massamongst nearby initial data (with the same interval structure) with fixedangular momenta. Thus the ADM mass of nearby data $m\geq f(J_1,J_2)$ for somefunction $f$ depending on the interval structure. The proof requires that theinitial data of the critical points satisfy certain conditions that aresatisfied by the extreme Myers-Perry and extreme black ring data.

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