The remaining classfication results assume cohomogeneity-two isometry actions :
Theorem 4.1 Let , , be two -regular, -dimensional, , stationary toroidal Kaluza–Klein spacetimes, with five asymptotically-large dimensions (). Assume, moreover, that the event horizon is connected and mean non-degenerate. If the interval structure and the set of angular momenta coincide, then the domains of outer communications are isometric.
This theorem generalizes previous results by the same authors [167, 168] as well as a uniqueness result for a connected spherical black hole of .
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)
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 .
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].
Living Rev. Relativity 15, (2012), 7
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