  We investigate the interior hyperbolic region of axisymmetric and stationary
black holes surrounded by a matter distribution. First, we treat the
corresponding initial value problem of the hyperbolic Einstein equations
numerically in terms of a single-domain fully pseudo-spectral scheme.
Thereafter, a rigorous mathematical approach is given, in which soliton methods
are utilized to derive an explicit relation between the event horizon and an
inner Cauchy horizon. This horizon arises as the boundary of the future domain
of dependence of the event horizon. Our numerical studies provide strong
evidence for the validity of the universal relation $\Ap\Am = (8\pi J)^2$ where
$\Ap$ and $\Am$ are the areas of event and inner Cauchy horizon respectively,
and $J$ denotes the angular momentum. With our analytical considerations we are
able to prove this relation rigorously.
