  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.
