proof of the global inequality (\[eq:60\]) was provided in a series of articles [@Dain05c], [@Dain05d], [@Dain05e] which end up in the global proof given in [@Dain06c]. The proof is based on a variational characterization of the extreme Kerr initial data. In [@Chrusciel:2007dd] and [@Chrusciel:2007ak] the result was generalized and the proof simplified. In [@Chrusciel:2009ki] [@Costa:2009hn] the charge was included. In [@Schoen:2012nh] relevant improvements on the rigidity statements were made. In particular in that article it was proved the first rigidity result including charge and a measure of the distance to extreme Kerr black hole was introduced. In [@zhou12] the result was proved with the maximal condition replaced by a small trace assumption for the second fundamental form of the initial data. Related results concerning the force between black holes were proved in [@Clement:2012np]. Finally, the mass formula and the variational techniques involved in the proof of the inequality (\[eq:60\]) were very recently used to study the linear stability of the extreme Kerr black hole [@Dain:2014iba].

Under the hypothesis of theorem \[t:1\] (namely, vacuum and axial symmetry) the angular momentum is defined as conserved quasi-local integral. In particular, if the topology of the manifold is trivial (i.e. ${\mathbb{R}^3}$), then the angular 