Jerzy Lewandowski: Isolated horizons, the Petrov type D equation and the Near Horizon Geometry equation

3-dimensional null surfaces that are Killing horizons to the
second order are considered. They are embedded in 4-dimensional
spacetimes that satisfy the vacuum Einstein equations with
arbitrary cosmological constant. Internal geometry of 2-dimensional
cross sections of the horizons consists of induced metric tensor
and a rotation 1-form potential. It is subject to the type D equation.
The equation is interesting from both the mathematical and physical
points of view. Mathematically it involves geometry, holomorphic
structures and algebraic topology. Physically, the equation knows the
secrete of black holes: the only axisymmetric solutions on topological
sphere correspond to the Kerr / Kerr-de Sitter /Kerr-anti-de-Sitter
non-extremal black holes or to the near horizon limit of the extremal
ones. In the case of bifurcated horizons the type D equation
implies another spacial symmetry. In this way the axial symmetry
may be ensured without the rigidity theorem. The type D equation
does not allow rotating horizons of topology different than that of
the sphere (or its quotient). That completes a new local
no-hair theorem. The type D equation is also an integrability condition
for the Near Horizon Geometry equation and leads to new results on
the solution existence issue.