Connected smooth axisymmetric vacuum Kerr classification

In the four-dimensional classical vacuum Einstein model with zero cosmological constant, assume the smooth global stationary-axisymmetric comparison data: a selected asymptotically flat end, complete stationary flow, global commuting axial circle action with regular axis, exact $I^{+}$-regularity and connected event horizon. These are the predicates of the supplied global axisymmetric comparison-data definition and black-hole comparison-hypothesis definition. Retain the mean-nondegeneracy hypothesis in the supplied Gaussian-null sense of the former definition. No additional orbit-space hypothesis is imposed beyond this explicit package. Work in ZFC, including choice. The model equation is $\mathrm{Ric}(g)=0$, the vacuum specialization of the adopted Einstein field law. Metric and causal notation use the stationary Lorentzian and causal-exterior interfaces.

The Kerr comparison object means the following full black-hole exterior family, including the extremal case. For $m>0$ and $\mathrm{\mid }a\mathrm{\mid }\le m$, set $r_{+}=m+\sqrt{m^2-a^2}$ and $D_{m,a}={\mathbb{R}}_t\times (r_{+},\mathrm{\infty }{)}_r\times S^2$, with its product smooth structure. Use the round sphere metric $\gamma$, the rotation-dual one-form $\lambda$ and the height function $\zeta$, pulled back to this product, in the declared Kerr tensor convention; thus away from the poles $\gamma =d{\theta }^2+{\sin }^2\theta d{\varphi }^2$, $\lambda ={\sin }^2\theta d\varphi$ and $\zeta =\cos \theta$. Put $\mathrm{\Sigma }=r^2+a^2{\zeta }^2$ and $\mathrm{\Delta }=r^2-2mr+a^2$. The comparison metric is the globally smooth sphere-tensor expression

$g_{m,a}=-(1-\frac{2mr}{\mathrm{\Sigma }})dt\otimes dt+\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}dr\otimes dr+\mathrm{\Sigma }\gamma +\frac{a^2(\mathrm{\Sigma }+2mr)}{\mathrm{\Sigma }}\lambda \otimes \lambda -\frac{2amr}{\mathrm{\Sigma }}(dt\otimes \lambda +\lambda \otimes dt)\mathrm{.}$

The independent Kerr geometry inputs identify $(D_{m,a},g_{m,a})$ as the open domain of outer communications relative to its selected asymptotically flat end, including $\mathrm{\mid }a\mathrm{\mid }=m$, when $r_{+}=m$ and $\mathrm{\Delta }=(r-m{)}^2$. The rotation axis is included through $S^2$; the event-horizon boundary $r=r_{+}$ is excluded. For $\mathrm{\mid }a\mathrm{\mid }<m$ this agrees with the supplied subextremal Kerr tensor datum. Reversal of azimuth gives the equivalent convention with $a$ replaced by $-a$.

Then for some such $m,a$ there exists a diffeomorphism $F:D\to D_{m,a}$ with $F^{\ast }{g}_{m,a}=g{\mathrm{\mid }}_D$. Thus the entire domain of outer communications $D$ is isometric to the domain of outer communications of a Kerr spacetime. The conclusion concerns the whole exterior, not just an Ernst potential or a neighborhood of the horizon, and does not classify arbitrary extensions beyond it.

No independent analyticity assumption is required for this axisymmetric comparison. Axisymmetry is already a global hypothesis; extending a locally constructed axial Killing field to all of $D$ is a separate obligation. The smooth classification source does not require mean-nondegeneracy, so this application retains that hypothesis without inferring pointwise nondegeneracy from it.

Smooth global data for the axisymmetric comparison

Stationary Lorentzian data and tensor conventions

Staticity, axisymmetry and black-hole regularity hypotheses

Causal exterior, chronological boundaries and Cauchy data

Adopted Einstein field equation in the direct-equation presentation

Subextremal Kerr exterior in length coordinates

The Axiom of Choice

Import the vacuum specialization of Chruściel–Costa–Heusler Theorem 3.2, read with §3.2.4's vacuum Kerr case. Theorem 3.2 gives an isometry of the full domain of outer communications; its electrovacuum conclusion is Kerr–Newman for a connected horizon, and the vacuum specialization is Kerr. Section 3.3.1 expressly distinguishes this theorem, requiring no analyticity, from the rigidity step that derives axisymmetry. Its explicit hypotheses are stationarity, axisymmetry, asymptotic flatness, $I^{+}$-regularity, dimension four and the electrovacuum equations, specialized here to vacuum; connectedness gives the Kerr conclusion. No unnamed additional orbit-space assumption is retained. Global orbit-space control, harmonic-map uniqueness and reconstruction are parts of this imported classification, not consequences proved by the elementary local suppliers.

Import the Kerr exterior manifold and its identification as the domain of outer communications for both subextremal and extremal parameters from Sbierski §3.2 and the opening of §3.2.1. His convention $0\le a\le m$, $m>0$, covers both regimes; reversing azimuth gives the negative-$a$ convention. This input identifies the full product exterior, including the sphere axis and excluding the horizon boundary; it is not merely a metric formula on a pole-free patch.

Import the metric coefficients and smooth spherical extension from Aretakis §§2.1–2.2, with the existing supplier's sphere-tensor notation. Section 2.1 fixes the Boyer–Lindquist coefficients in the $(-,+,+,+)$ convention; §2.2 supplies the regular Lorentzian metric and explicitly identifies the extremal domain of outer communications as $r>m$. Thus extremality is included without extrapolating the published subextremal interface's hypotheses. The whole-exterior classification and these comparison-family geometry facts are external established inputs; no local proof of them is claimed.

Chruściel–Costa 0806.0016 Theorem 1.3 is not invoked as a smooth theorem, and its Corollary 6.3 alone is not treated as a full metric classification. The five local intermediate items define the exact global hypothesis package and the elementary action, orbit and reconstruction algebra. They do not reconstruct the imported global classification proof or assert a mean-to-pointwise nondegeneracy bridge. Exact source hypotheses and actual reading limits appear in the references.