Considering two arbitrary solutions of the Ernst equations, Robinson was able to construct an identity , the integration of which proved the uniqueness of the Kerr metric. The complicated nature of the Robinson identity dashed the hope of finding the corresponding electrovacuum identity by trial and error methods (see, e.g., ). The problem was eventually solved when Mazur [228, 230] and Bunting  independently derived divergence identities useful for the problem at hand. Bunting’s approach, applying to a general class of harmonic mappings between Riemannian manifolds, yields an identity, which enables one to establish the uniqueness of a harmonic map if the target manifold has negative curvature. We refer the reader to Sections 3.2.5 and 8.4.2 (see also ) for discussions related to Bunting’s method.
So, consider two solutions of the Ernst equations associated to, a priori, distinct black-hole spacetimes, each endowed with its own metric. As discussed in Section 8.3, Weyl coordinates and conformal invariance allow us to view the Ernst equations as equations on a flat half-plane; alternatively, they may be seen as equations for an axisymmetric field on three-dimensional flat space. The Mazur identity (7.2) applies to the relative difference of the associated Hermitian matrices and implies (see Section 7.1 for details and references)
The reduction of the EM equations with respect to the axial Killing field yields -model equations with target (see Section 6.4), in vacuum reduces to (see Section 6.2). Hence, the above formula applies to both the stationary and axisymmetric vacuum or electrovacuum field equations. Now, relying on axisymmetry once more, we can reduce the previous Mazur identity to an equation on the flat half-plane ; integrating and using Stokes’ theorem leads to
The uniqueness of the Kerr–Newman metric should follow now from
In order to establish that on the boundary of the half-plane,11 one needs the asymptotic behavior and the boundary and regularity conditions of all potentials. One expects that vanishes on the horizon, the axis and at infinity, provided that the solutions have the same mass, charge and angular momentum, but no complete analysis of this has been presented in the literature; see  for some partial results. Fortunately, the supplementary difficulties arising from the need to control the derivatives of the fields disappear when the distance-function approach described in the next Section 8.4.2 is used.
An alternative to the divergence identities above is provided by the observation that the distance between two harmonic maps , , with negatively curved target manifold is subharmonic [182, Lemma 8.7.3 and Corollary 8.6.4] (see also the proof of Lemma 2 in  following results in ):[144, Theorem 3.1] after noting that the target spaces of the maps under consideration are of non-compact type (see also ).
Using this observation, the key to uniqueness is provided by the following non-standard version of the maximum principle:
Proposition 8.1 [75, Appendix C] Let denote the -axis in , and let satisfyand Then
Hence, to prove uniqueness it remains to verify that is bounded on , and that
Living Rev. Relativity 15, (2012), 7
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