Local Kerr rigidity and conditional global identification

Apply [F1] using the explicit maximality hypothesis on the actual exterior section $C={\mathrm{\Sigma }}_0\cap E$. The conditional lapse/product conclusion gives positive stationary lapse, its equation and zero boundary value, and a complete Killing-flow product. Its intrinsic closure is smooth, complete and spherical; no agreement of an auxiliary global radial chart with the physical end is asserted. On the original physical end, the lapse expansion in the mass-flux lemma gives $N=1-M\mathrm{/}r+O_1(r^{-2})$. The compact-core and physical-end conclusions of [F6] verify its exterior hypothesis, so that lemma gives $M>0$.

By [F6] choose an adapted topological identification $j:{\mathrm{\Sigma }}_0\to \{\mathrm{\mid }z\mathrm{\mid }>1\mathrm{/}2\}$ with $j(S)=\{\mathrm{\mid }z\mathrm{\mid }=1\}$ and $j(C)=\{\mathrm{\mid }z\mathrm{\mid }>1\}$. It need not agree with the physical end chart $x$. Choose a smooth nonincreasing function $\chi$ equal to one near $1\mathrm{/}2$ and zero for $u\ge 3\mathrm{/}4$. For example, normalize the integral from $u$ to $3\mathrm{/}4$ of the positive bump $\exp (-1\mathrm{/}(v-5\mathrm{/}8)-1\mathrm{/}(3\mathrm{/}4-v))$ on $5\mathrm{/}8<v<3\mathrm{/}4$, extended by zero, and extend the resulting cutoff constantly at either end. The function $a(u)=u-\chi (u)\mathrm{/}2$ has $a^{\mathrm{\prime }}\ge 1$, endpoint limits zero and infinity, and equals $u$ for $u\ge 3\mathrm{/}4$. Thus $z=a(\mathrm{\mid }j\mathrm{\mid })j\mathrm{/}\mathrm{\mid }j\mathrm{\mid }$ is a diffeomorphism from ${\mathrm{\Sigma }}_0$ to punctured ${\mathbb{R}}^3$, unchanged near $S$ and on $C$. Add the point $p_0$ at $z=0$ to obtain an auxiliary smooth manifold $W\cong {\mathbb{R}}^3$, and let $D_0=\{\mathrm{\mid }z\mathrm{\mid }\le 1\}$. This is a parametrized smooth closed ball with center $p_0$ and boundary $S$. This auxiliary filling is only a spatial-coordinate construction; it adds no point to the spacetime or its metric.

Choose a sufficiently large physical radius $R>2$ so that the compact physical truncation of [F6] is defined and a two-sided coordinate collar of ${\mathrm{\Gamma }}_R=\{t=0,\mathrm{\mid }x\mathrm{\mid }=R\}$ lies in the physical end. Set $V_R=\{t=0,\mathrm{\mid }x\mathrm{\mid }>R\}$ and $D_R=W\setminus V_R$. The compact-truncation contract makes $D_R$ compact, with only boundary ${\mathrm{\Gamma }}_R$, and $D_0\subset \mathrm{int}{D}_R$. The tail $V_R$ is connected and cofinal in the unique end, hence is unbounded in $z$. The smooth sphere theorem in [F6], applied to ${\mathrm{\Gamma }}_R$ in the one-point compactification of $W\cong {\mathbb{R}}^3$, gives precisely two complementary components with smooth ball closures. Since $V_R$ is connected, has boundary exactly ${\mathrm{\Gamma }}_R$ and is unbounded, it is the component containing the compactifying point after that point is removed. Therefore $D_R$ is the bounded-side smooth standard closed three-ball.

Take a smooth ball diffeomorphism $\beta :D_R\to {\bar{B}}_R(0)$. The boundary map $h=x\circ {\beta }^{-1}:\mathrm{\partial }{B}_R\to \mathrm{\partial }{B}_R$ is a smooth diffeomorphism. The prescribed sphere-map extension in [F7], after constant rescaling, gives a smooth ball diffeomorphism $\bar{h}$ extending $h$, so $g_1=\bar{h}\circ \beta$ agrees with $x$ on ${\mathrm{\Gamma }}_R$. To obtain agreement of germs, choose small $ϵ>0$ and, for $p\in {\mathrm{\Gamma }}_R$ with $x(p)=R\omega$, form the two closed inward collars $c_0(p,s)=x^{-1}((R-ϵs)\omega )$ and $c_1(p,s)=g_1^{-1}((R-ϵs)\omega )$, $0\le s\le 1$. Both are smooth embeddings and have boundary value $p$. Closed-collar uniqueness in [F7] gives a diffeomorphism $k$ of $D_R$, fixing its boundary, with $k\circ c_0=c_1$. Thus $g=g_1\circ k$ equals $x$ on the whole physical inward collar. Gluing $g$ on $D_R$ to $x$ on $V_R$ now yields a smooth global diffeomorphism $G:W\to {\mathbb{R}}^3$, since both formulas equal the same physical coordinates near the seam. It is exactly $x$ on the far tail.

Let $e_0:{\bar{D}}^3\to D_0$ be the parametrization from $z$, with $e_0(0)=p_0$. The embeddings $f=G\circ e_0$ and $e(v)=v$ of ${\bar{D}}^3$ have image in the connected boundaryless manifold $\mathrm{int}{B}_R$. The disk theorem in [F7] gives an isotopy of $f$ to $e$ or to $e\circ s$, where $s$ is a source reflection fixing zero. Its compact source has compact track. Apply [F8] to extend that isotopy with support in a compact subset of $\mathrm{int}{B}_R$, then extend its terminal diffeomorphism $H$ by the identity to all of ${\mathbb{R}}^3$. Thus $H(G(D_0))={\bar{B}}_1$ and $H(G(p_0))=0$, while $H$ is the identity near and outside $\mathrm{\partial }{B}_R$. The radial diffeomorphism $A(v)=a^{-1}(\mathrm{\mid }v\mathrm{\mid })v\mathrm{/}\mathrm{\mid }v\mathrm{\mid }$ from punctured ${\mathbb{R}}^3$ to $\{\mathrm{\mid }X\mathrm{\mid }>1\mathrm{/}2\}$ is the identity for $\mathrm{\mid }v\mathrm{\mid }\ge 3\mathrm{/}4$. Hence $X=A\circ H\circ G{\mathrm{\mid }}_{{\mathrm{\Sigma }}_0}$ is a global smooth section chart with $X(S)=\{\mathrm{\mid }X\mathrm{\mid }=1\}$, $X(C)=\{\mathrm{\mid }X\mathrm{\mid }>1\}$ and $X=x$ on the physical tail $\mathrm{\mid }x\mathrm{\mid }>R$. The possible reflection changes only the cap parametrization, not the fixed physical end coordinates. No section, metric or maximality property was changed. Transporting $X$ by the stationary flow gives spatial parameters on the product. Its time vanishes on $C$ and has derivative one along $T$, so on the original end it is the original stationary time. This establishes the full global-chart agreement required by the cutoff supplier before its application.

Match IK AF on the original ambient spacetime: the chart $X$ of step 4.1 agrees with the original physical spatial coordinates at infinity and puts $S$ at radius one; the original $O_6$ bounds imply IK's undifferentiated AF bounds; $T={\mathrm{\partial }}_t$, $M>0$ by step 1.1, and the complete orbits meet $C$. Its future normal satisfies $\mathrm{\mid }g(T,n)\mathrm{\mid }=N>0$ there. The actual smooth transverse null nonexpanding boundary branches, their sphere topology and tangency with $T{\mathrm{\mid }}_S\equiv ̸0$ are precisely SBS. The same supplied Ernst potential tends uniformly to one along the physical $t=0$ end; both assumed sphere conditions are exactly IK T in the fixed orientation and curvature conventions. Thus the local-criterion interface gives exactly local isometry to a Kerr DOC of mass $M$ with $0<a_{\mathrm{l}\mathrm{o}\mathrm{c}}<M$. Apply the separate reciprocal-invariant supplier under these same matched AF/SBS/T hypotheses to obtain $1-\sigma \mathrm{\ne }0$, $\mathcal{S}=0$ and a single $0\le B<1\mathrm{/}4$ on $E$, with its stated horizon values. The tensor/scalar assertions are not inferred from the local-isometry conclusion alone.

Import the stationary-end compactification/analyticity inputs through the normalization lemma. It applies on the timelike far tail only, with the original O6 bounds and positive mass just established, and gives the exact AIK O6 angular-momentum chart. In this normalization, the old and reconstructed stationary connection one-forms have the same exterior derivative on the spherical tail. Their difference is exact by the spherical-period input, producing the exact stationary time change. The same $T$ and normalized Ernst fields are used. Alternation gives $d\sigma (T)=2\mathcal{F}(T,T)=0$, so $y,z$ are stationary. The normalized metric bounds give $g^{ij}={\delta }^{ij}+O(r^{-1})$ by inversion near the identity. The normalized asymptotic expansion and source Appendix A yield $y=r\mathrm{/}(2M)+O(1)$, $z=(\mathbf{S}\cdot x)\mathrm{/}(2M^{2}r)+O_1(r^{-1})$. Apply the scalar-spin lemma to the single constant B of step 5.1, obtaining $B=J^2\mathrm{/}(4M^4)$ and $J=\mathrm{\mid }\mathbf{S}\mathrm{\mid }<M^2$. The normalization lemma's ADM integral identifies this coefficient with physical ADM spin, rather than an unrelated source symbol. This uses neither the desired global isometry nor AIK's assumed parameter bound to prove that bound.

The horizon scalar relations in [F2] exclude $B=0$. If it were zero, $y{\mathrm{\mid }}_S=1$ and $z{\mathrm{\mid }}_S=0$, hence $\sigma {\mathrm{\mid }}_S=0$. The vacuum Ernst identity and common end normalization give $g(T,T){\mathrm{\mid }}_S=0$. Since $T$ is tangent to both transverse horizon branches, it is tangent to their spacelike intersection $S$; its positive definite induced metric then forces $T=0$ everywhere on $S$, contradicting the rotating hypothesis. Thus $0<B<1\mathrm{/}4$, and step 6.1 gives $0<J<M^2$. This conclusion precedes, and does not assume, a global Kerr identification.

Use the global section chart $X$ of step 4.1 in the cutoff supplier; it places $S$ at unit radius and agrees exactly with the original physical coordinates on a far tail. Choose the cutoff radius beyond that tail threshold. The stationary product time is Cauchy temporal by its supplier. Use the coordinate/time cutoff lemma to extend the normalized end chart while keeping the section unchanged near S; its bounded spacelike graph C-prime is Cauchy in E and has the original spherical-exterior topology. New-slice maximality is unnecessary. Now impose the Statement's additional access hypothesis on the selected stationary end $U_f\subset U_0\cap E$, chosen beyond all cutoff thresholds. The two identities $I_E^{-}(U_f)=I_E^{+}(U_f)=E$ give both interior access statements directly. Inclusion of these curves in $M$ and $U_f\subset U_0$ give $I_M^{-}(U_f)\cap I_M^{+}(U_f)=E$ by the two opposite inclusions; ambient DOC equality is not used to infer interior access. The working-bifurcation-collar lemma gives an open spacetime M-prime containing E and S with a smooth embedded partial Cauchy extension Sigma-prime and both separate source equalities Sigma-prime intersect I-minus(end)=Sigma-prime intersect I-plus(end)=C-prime. Its global spatial chart agrees with the normalized chart at infinity and places S at radius1. More precisely, write $U_f$ for this end and $O$ for the collar in the strengthened working-collar interface. It gives separately $O\cap I_{M^{\mathrm{\prime }}}^{-}(U_f)=O\cap I_M^{-}(U_0)$ and $O\cap I_{M^{\mathrm{\prime }}}^{+}(U_f)=O\cap I_M^{+}(U_0)$, hence $O\cap {\mathrm{\partial }}_{M^{\mathrm{\prime }}}{I}_{M^{\mathrm{\prime }}}^{-}(U_f)=O\cap {\mathrm{\partial }}_M{I}_M^{-}(U_0)$ and $O\cap {\mathrm{\partial }}_{M^{\mathrm{\prime }}}{I}_{M^{\mathrm{\prime }}}^{+}(U_f)=O\cap {\mathrm{\partial }}_M{I}_M^{+}(U_0)$. Their full double boundary in $M^{\mathrm{\prime }}$ is $S$. The supplier proves these individual identities by timelike curves from the old past set to $E$ and from $E$ to the old future set within $O$, followed by two-sided tail access. Thus the actual new boundaries required by AIK SBS are the old smooth transverse null branches near $S$. The metric and its connection are restricted without alteration, so their null second fundamental forms and zero expansions are unchanged; $T$ remains tangent to both and is not identically zero on $S$. These are the retained horizon and section clauses in the conditional global matching; the complete source-native package is still imposed separately in the final application.

For the global implication, now assume the entire additional AIK GR/SBS/PK package for the working tuple specified in the Statement, including every source regularity/neighborhood condition defining finite $A$. Step 7.2 identifies its DOC with the original $E$ and preserves the two individual horizon boundaries and chronological-section equalities. The normalized chart, mass and spin in step 6.1 are the same data as in GR, with the positive subextremal range proved in step 7.1. Positive lapse and stationary transport make $T$ nonzero in $E$, and the bounded-graph supplier makes every complete orbit meet $C^{\mathrm{\prime }}$ once. The normalized-field lemma preserves the same Ernst potential and geometric Mars–Simon tensor under the passive chart/time changes, so its zero value from step 5.1 gives zero in every source frame contraction in PK for every positive threshold. This observation does not discharge the other source clauses, which remain explicit matching hypotheses. Apply the independent AIK Main Theorem under that whole package to obtain one global isometry from $E$ onto the full Kerr DOC of mass $M$ and ADM angular momentum magnitude $J$. With the Kerr convention this is the full exterior with $J=M\mathrm{\mid }a\mathrm{\mid }$ and $0<\mathrm{\mid }a\mathrm{\mid }<M$; it asserts no extension across its horizon. The conditional composition proves the stated global implication without claiming the original theorem under its shorter assumptions.