Axisymmetric vacuum uniqueness with complete global hypotheses

PDF p. 2: §1.1, Carter–Bunting/Robinson discussion

Adopt the smooth four-dimensional classical stationary vacuum model $\mathrm{Ric}(g)=0$ with zero cosmological constant. Assume the exact smooth global axisymmetric comparison data relative to the selected asymptotically flat end: the stationary Killing flow is complete on $M$ and timelike near that end; there is a global effective isometric axial circle action of period $2\pi$ commuting with that flow and with the specified regular axis; the entire exterior $D=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ is $I^{+}$-regular in the full stated sense; and its event horizon is connected. Then $D$, with its metric, is isometric to the domain of outer communications of a Kerr spacetime in the convention of the Kerr definition.

No analyticity is required for this already axisymmetric classification. It concerns the entire exterior, not merely a neighborhood of the horizon or a reduced potential, and asserts no classification of arbitrary extensions beyond the exterior. A global axial action is a hypothesis: existence of a local horizon Killing field alone does not satisfy it. If mean-nondegeneracy or nondegeneracy is also imposed in an application, retain that separate hypothesis; the imported axisymmetric theorem does not need it, and no mean-to-pointwise nondegeneracy inference is made. The complete source theorem imposes no additional unspecified orbit-space hypotheses beyond its stated global assumptions.

Imported established classification: Theorem3.2: a stationary, axisymmetric, asymptotically flat, $I^{+}$-regular electrovacuum four-dimensional spacetime has domain of outer communications isometric to a Weinstein solution; if the event horizon is connected, it is the domain of outer communications of Kerr–Newman. The vacuum case is the Kerr uniqueness conclusion in §3.2.4. Section3.3.1 states explicitly that this axisymmetric theorem requires no analyticity. Hypotheses: Four-dimensional smooth spacetime; complete stationary Killing flow timelike at the selected asymptotically flat end; global stationary-axisymmetric structure, not a merely local Killing field; full I-plus-regularity of Definition2.1; vacuum specialization and connected event horizon. The local item assumes an effective commuting axial circle action with regular axis explicitly. Nondegeneracy is not a hypothesis of Theorem3.2. There is no additional unspecified orbit-space assumption in its stated theorem; the global reduction and reconstruction are part of the imported classification. Reading limits: Read the specified definitions, theorem statement and adjacent discussions from the downloaded full PDF, visually inspected p.15. Did not read or reconstruct the cited primary Robinson, Carter, Bunting, Weinstein or global orbit-space proofs. The full exterior classification is accepted as an external established input; the five local helpers prove only elementary prerequisite bridges. The source itself is a review theorem, not a newly reconstructed proof.

Original article attribution: The uniqueness of smooth stationary black holes, arXiv:2611.16853v1, §1.1, PDF p.2, Carter–Bunting/Robinson discussion. That paragraph reports the stationary-axisymmetric Kerr conclusion and says its final uniqueness step does not require analyticity. It is retained as attribution only; the independent CCH theorem supplies the classification. The ambiguous generated supplier phrase about additional unspecified orbit-space hypotheses is not used as a premise. Its original item and authentic history remain untouched. The five local prerequisite lemmas establish only the stated elementary bridges; no orbit-space topology, harmonic-map uniqueness or global reconstruction is claimed locally.