Reported analytic and smooth electrovacuum uniqueness literature

PDF p. 3: §1.1, final paragraph

The article mentions uniqueness results for stationary electrovacuum black holes in the analytic case, citing its references [8, 13, 30], and in the smooth case, citing [28, 42]. The passage does not give a complete common theorem or hypothesis list for those works. In particular, this report does not assert that every smooth stationary electrovacuum black hole is uniquely determined without further hypotheses.

For the terminology used here, source-free geometric electrovacuum data comprise a smooth time-oriented and oriented Lorentzian four-manifold with a smooth real two-form $F$ satisfying $dF=0$, $d({\star }_{g}F)=0$, and ${\mathrm{Ric}}_{ab}=2(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})$, in the stated geometric normalization. In the asymptotically flat black-hole setting used here, stationarity relative to a selected asymptotically flat spacelike end means that a specified Killing field is future timelike on a sufficiently distant tail of that end. The local flow of a Killing field preserves the metric wherever defined. Analytic means that the metric is represented locally by convergent real power series in a compatible real-analytic atlas; smooth means $C^{\mathrm{\infty }}$. A uniqueness result identifies solutions within the class and under the hypotheses specified by that result. The precise horizon, regularity, field and asymptotic hypotheses must therefore be read separately for each cited result. The geometric suppliers use the Axiom of Choice.

Geometric electrovacuum equations and charge convention

Staticity, axisymmetry and black-hole regularity hypotheses

Killing fields of a Lorentzian metric

The Axiom of Choice

References [8] and [30] in the article are works by Bunting and Mazur; [13] is the review by Chruściel, Costa and Heusler. References [28] and [42] are works by Lai, Li and Yu, and by Yu, respectively. These are bibliographic identifications from the article, not independent verification of their theorem statements or proofs. The citation to the article supplies attribution for the report above; it is not a proof of electrovacuum uniqueness.