The near (extremal) horizon geometry equation

The Near Horizon Geometry (NHG) equation emerges in several contexts of general relativity and theoretical physics. In the quasi local theory of black holes, this equation is a constraint on the induced geometry of extremal isolated horizon.  It considerably restricts possible local  spacetime metric tensors that satisfy the Einstein equations and admit an extremal Killing horizon.  Another context is that of  null, non-expanding foliations. I that case a certain equation that  constrains possible geometries of the leaves of the foliation turns our to be technically equivalent  to the NHG equation. The third application of the equation is a construction of exact solutions to  Einstein’s equations that describe the vicinity  of the extremal black hole horizon. They have become  famous as background for the Kerr/CFT correspondence.  The history of the equation will be outlined  in the talk. New results on the NHG equation will be presented: robustness of the relation with the  null non-expanding foliations,  new  integrability conditions and their holomorphic interpretation, a  general solution to the NHG equation for horizons that admit 2d cross sections of  non-positive Euler  class.  The results concerning the topologically spherical sections of extremal horizon will be discussed.  That case is still open and is waiting for a solution.