## Extreme Black Holes and Near-Horizon Geometries

##### Abstract

In this thesis we study near-horizon geometries of extreme black holes. We first consider
stationary extreme black hole solutions to the Einstein-Yang-Mills theory with a
compact semi-simple gauge group in four dimensions, allowing for a negative cosmological
constant. We prove that any axisymmetric black hole of this kind possesses a
near-horizon AdS2 symmetry and deduce its near-horizon geometry must be that of the
abelian embedded extreme Kerr-Newman (AdS) black hole. We show that the near-horizon
geometry of any static black hole is a direct product of AdS2 and a constant
curvature space. We then consider near-horizon geometry in Einstein gravity coupled
to a Maxwell field and a massive complex scalar field, with a cosmological constant. We
prove that assuming non-zero coupling between the Maxwell and the scalar fields, there
exists no solution with a compact horizon in any dimensions where the massive scalar
is non-trivial. This result generalises to any scalar potential which is a monotonically
increasing function of the modulus of the complex scalar.
Next we determine the most general three-dimensional vacuum spacetime with a negative
cosmological constant containing a non-singular Killing horizon. We show that the
general solution with a spatially compact horizon possesses a second commuting Killing
field and deduce that it must be related to the BTZ black hole (or its near-horizon
geometry) by a diffeomorphism. We show there is a general class of asymptotically
AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry
Virasoro algebras and vanishing charges with respect to the other. We
interpret these as descendants of the extreme BTZ black hole. However descendants
of the non-extreme BTZ black hole are absent from our general solution with a non-degenerate
horizon.
We then show that the first order deformation along transverse null geodesics about any
near-horizon geometry with compact cross-sections always admits a finite-parameter
family of solutions as the most general solution. As an application, we consider the
first order expansion from the near-horizon geometry of the extreme Kerr black hole.
We uncover a local uniqueness theorem by demonstrating that the only possible black
hole solutions which admit a U(1) symmetry are gauge equivalent to the first order
expansion of the extreme Kerr solution itself. We then investigate the first order expansion
from the near-horizon geometry of the extreme self-dual Myers-Perry black
hole in 5D. The only solutions which inherit the enhanced SU(2) X U(1) symmetry
and are compatible with black holes correspond to the first order expansion of the extreme
self-dual Myers-Perry black hole itself and the extreme J = 0 Kaluza-Klein black
hole. These are the only known black holes to possess this near-horizon geometry. If
only U(1) X U(1) symmetry is assumed in first order, we find that the most general
solution is a three-parameter family which is more general than the two known black
hole solutions. This hints the possibility of the existence of new black holes.