### 4.6 Classification theorems for KK-black holes

As usual, the static case requires separate consideration. The first classification results addressed static
five-dimensional solutions with KK asymptotics and with a factor in the group of isometries. In
such a setting, the Kaluza–Klein reduction leads to gravity coupled with a Maxwell field and a
“dilaton” field , with a Lagrangean
where . In the literature one also considers more general theories where does not necessarily
take the Kaluza–Klein value. All current uniqueness proofs require that the mass, the Maxwell charges, and
the dilaton charge satisfy a certain genericity condition, and that all horizon components have
non-vanishing surface gravity. When , Mars and Simon [224] show that the generic static solutions
belong to the family found by Gibbons and Maeda [131, 126, 122]. For other values of , in
particular for the KK value, a purely electric or purely magnetic configuration is assumed, and
then the same conclusion is reached. The result is an improvement on the original uniqueness
theorems of Simon [294] and Masood-ul-Alam [226], and has been generalized to higher dimensions
in [129]. The analyticity assumption, which is implicit in all the above proofs, can be removed
using [72].
The remaining classfication results assume cohomogeneity-two isometry actions [169]:

This theorem generalizes previous results by the same authors [167, 168] as well as a uniqueness result
for a connected spherical black hole of [240].

The proof of Theorem 4.1 can be outlined as follows: After establishing the, mainly topological, results
of Sections 4.3, 4.4 and 4.5, the proof follows closely the arguments for uniqueness of 4-dimensional
stationary and axisymmetric electrovacuum black holes. First, a generalized Mazur identity is valid in
higher dimensions (see [218, 31] and Section 7.1). From this Hollands and Yazadjiev show that (compare
the discussion in Sections 8.4.1 and 8.4.2)

where is the flat Laplacian on , the function
is
defined as
the
’s, , are the Mazur matrices [169, Eq. (78)] associated with the two black-hole spacetimes
that are being compared. In terms of twist potentials and metric components of the axisymmetric
Killing vectors (generators of the toroidal symmetry)
we
have the following explicit formula (see also [218, 240])
where and where is the matrix inverse to .
It should be noted that this provides a variation on Mazur’s and the harmonic map methods (see
Sections 3.2.4 and 3.2.5), which avoids some of their intrinsic difficulties. Indeed, the integration by
parts argument based on the Mazur identity requires detailed knowledge of the maps under
consideration at the singular set , while the harmonic map approach requires finding, and
controlling, the distance function for the target manifold. (In some simple cases is the desired
distance function, but whether this is so in general is unclear.) The result then follows by a careful
analysis of the asymptotic behavior of the relevant fields; such analysis was also carried out
in [169].

In this context, the degenerate horizons suffer from the supplementary difficulty of controlling the
behavior of the fields near the horizon. One expects that an exhaustive analysis of near-horizon
geometries would allow one to settle the question; some partial results towards this can be found
in [204, 203, 202, 164].