Let us start with the first issue, concerning the generality of the strong rigidity theorem (SRT). The existence of a second Killing vector field to the future of a bifurcation surface can be established by solving a characteristic Cauchy problem , which makes it clear that axial symmetry will hold for a large class of matter models satisfying the, say, dominant energy condition.
The counterpart to the staticity problem is the circularity problem: As general non-rotating black holes are not static, one expects that the axisymmetric ones need not be circular. This is, indeed, the case: While circularity is a consequence of the EM equations and the symmetry properties of the electro-magnetic field, the same is not true for the EYM system. In the Abelian case, the proof rests on the fact that the field tensor satisfies , and being the stationary and the axial Killing field, respectively; for Yang–Mills fields these conditions do no longer follow from the field equations and their invariance properties (see Section 8.1 for details). Hence, the familiar Papapetrou ansatz for a stationary and axisymmetric metric is too restrictive to take care of all stationary and axisymmetric degrees of freedom of the EYM system. However, there are other matter models for which the Papapetrou metric is sufficiently general: the proof of the circularity theorem for self-gravitating scalar fields is, for instance, straightforward . Recalling the key simplifications of the EM equations arising from the (2+2)-splitting of the metric in the Abelian case, an investigation of non-circular EYM equations is expected to be rather awkward. As rotating black holes with hair are most likely to occur already in the circular sector (see the next paragraph), a systematic investigation of the EYM equations with circular constraints is needed as well.
The static subclass of the circular sector was investigated in studies by Kleihaus and Kunz (see  for a compilation of the results). Since, in general, staticity does not imply spherical symmetry, there is a possibility for a static branch of axisymmetric black holes without spherical symmetry. Using numerical methods, Kleihaus and Kunz have constructed black-hole solutions of this kind for both the EYM and the EYM-dilaton system . The related axisymmetric soliton solutions without spherical symmetry were previously obtained by the same authors [190, 191]; see also  for more details. The new configurations are purely magnetic and parameterized by their winding number and the node number of the relevant gauge field amplitude. In the formal limit of infinite node number, the EYM black holes approach the Reissner–Nordström solution, while the EYM-dilaton black holes tend to the Gibbons–Maeda black hole [126, 131]. The solutions themselves are neutral and not spherically symmetric; however, their limiting configurations are charged and spherically symmetric. Both the soliton and the black-hole solutions of Kleihaus and Kunz are unstable and may, therefore, be regarded as gravitating sphalerons and black holes inside sphalerons, respectively.
Existence of slowly rotating regular black-hole solutions to the EYM equations was established in . Using the reduction of the EYM action in the presence of a stationary symmetry reveals that the perturbations giving rise to non-vanishing angular momentum are governed by a self-adjoint system of equations for a set of gauge invariant fluctuations . With a soliton background, the solutions to the perturbation equations describe charged, rotating excitations of the Bartnik–McKinnon solitons . In the black-hole case the excitations are combinations of two branches of stationary perturbations: The first branch comprises charged black holes with vanishing angular momentum,8 whereas the second one consists of neutral black holes with non-vanishing angular momentum. (A particular combination of the charged and the rotating branch was found in .) In the presence of bosonic matter, such as Higgs fields, the slowly rotating solitons cease to exist, and the two branches of black-hole excitations merge to a single one with a prescribed relation between charge and angular momentum . More information about the EYM–Higgs system can be found in [209, 254].
Living Rev. Relativity 15, (2012), 7
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