Proof of the angular momentum-mass inequality for axisymmetric black holes

We prove that extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric, black holes. As a consequence, we obtain that any data in this class satisfy the inequality $\sqrt{J}\le m$, where $m$ and $J$ are the total mass and angular momentum of the spacetime.