Mysteries of isolated horizons

Mysteries of isolated horizons: the Near Horizon Geometry equation, geometric characterizations of the non-extremal Kerr horizon, spacetimes foliated by non-expanding horizons.

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 the both, 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 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 then that of the  sphere (or its quotient). That completes a new local non-her 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.