### 2.8 Uniqueness of stationary near-horizon geometries

We reviewed in Section 2.3 that for any stationary extremal spinning black hole one can isolate a
geometry in the vicinity of the horizon, which has enhanced symmetry and universal properties. We
discussed in Section 2.7 that another class of stationary near-horizon geometries can be defined, which are,
however, related to the extremal near-horizon geometries via a diffeomorphism. It is natural to ask how
unique the stationary near-horizon geometries are.
In the case of Einstein gravity, one can prove that the NHEK (near-horizon extremal Kerr geometry) is
the unique (up to diffeomorphisms) regular stationary and axisymmetric solution asymptotic to the
NHEK geometry with a smooth horizon [4]. This can be understood as a Birkoff theorem for
the NHEK geometry. This can be paraphrased by the statement that there are no black holes
“inside” of the NHEK geometry. One can also prove that there is a near-horizon geometry
in the class (25), which is the unique (up to diffeomorphisms) near-horizon stationary and
axisymmetric solution of AdS–Einstein–Maxwell theory [192, 193, 191]. The assumption of
axisymmetry can be further relaxed since stationarity implies axisymmetry [170]. It is then natural to
conjecture that any stationary solution of the more general action (1), which asymptotes to a
near-horizon geometry of the form (25) is diffeomorphic to it. This conjecture remains to be
proven.