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