  We investigate the structure of the steady-state force-free magnetosphere
around a Kerr black hole in various astrophysical settings. The solution
Psi(r,theta) depends on the distributions of the magnetic field line angular
velocity omega(Psi) and the poloidal electric current I(Psi). These are
obtained self-consistently as eigenfunctions that allow the solution to
smoothly cross the two singular surfaces of the problem, the Inner Light
Surface (ILS) inside the ergosphere, and the Outer Light Surface (OLS), which
is the generalization of the pulsar light cylinder. Magnetic field
configurations that cross both singular surfaces (e.g. monopole, paraboloidal)
are uniquely determined. Configurations that cross only one light surface e.g.
the artificial case of a rotating black hole embedded in a vertical magnetic
field) are degenerate. We show that, similarly to pulsars, black hole
magnetospheres naturally develop an electric current sheet that potentially
plays a very important role in the dissipation of black hole rotational energy
and in the emission of high-energy radiation.
