The extremity assumptions imposes a set of non-trivial equations on the intrinsic and extrinsic geometry of the horizon. Some of them already got more attention since they are applied in the near horizon geometries (NHG), some other are less know. All of them are very important for the existence and uniqueness of extremal black hole solutions to Einstein's equations. In 4d spacetime, an integrability condition for the NHG equation is available that is satisfied by the family of non-extremal horizon geometries, namely by those that are of the Petrov type D. What is special about that equation, is that its solutions exhibit similar properties to those proved in the global black hole spacetime theory: spherical topology of horizon cross sections, rigidity, no-hair.
Extension of the research to the horizons that had a Hopf bundle
structure led to interesting results about the global structure of the
Kerr-NUT-(A)dS spaces. Misner's construction was generalised to the case
when none of the three parameters (NUT, Kerr, and the cosmological
constant) vanishes. The resulting family contains also exact solutions
to Einstein's equations that admit no horizon or nowhere time like
Killing vector and describe a topologically spherical universe evolving
from the past scry to the future scry in a non-singular manner.