  Recently it was shown that the area A and the angular momentum J of any
apparent horizon on a maximal, axisymmetric and asymptotically flat Cauchy
hyper-surface of a vacuum space-time satisfy necessarily the universal
inequality A >= 8 pi |J|. We show here that the equality A=8 pi |J| is never
attained. As equality is reached (on globally different data sets) when and
only when the surface is an extreme Kerr-throat sphere which has zero
"temperature", then our statement could be rephrased following this
thermodynamic heuristic as the non-existence of apparent horizons of zero
temperature. We study too the global structure of data sets having surfaces
with A=8 pi |J|. This lead us to prove the rigidity of the extreme Kerr-throats
and to investigate the important phenomenon of formation of extreme
Kerr-throats along sequences of data sets.
