  We give a thorough description of the shape of rotating axisymmetric stable
black-hole (apparent) horizons applicable in dynamical or stationary regimes.
It is found that rotation manifests in the widening of their central regions
(rotational thickening), limits their global shapes to the extent that stable
holes of a given area A and angular momentum J (non zero) form a precompact
family (rotational stabilization) and enforces their whole geometry to be close
to the extreme-Kerr horizon geometry at almost maximal rotational speed
(enforced shaping). The results, which are based on the stability inequality,
depend only on A and J. In particular they are entirely independent of the
surrounding geometry of the space-time and of the presence of matter satisfying
the strong energy condition. A complete set of relations between A, J, the
length L of the meridians and the length R of the greatest axisymmetric circle,
is given. We also provide concrete estimations for the distance between the
geometry of horizons and that of the extreme Kerr, in terms only of A and J.
Besides it own interest, the work has applications to the Hoop conjecture as
formulated by Gibbons in terms of the Birkhoff invariant, to the Bekenstein-Hod
entropy bounds and to study the compactness of classes of stationary black-hole
space-times.
