Rotating Kerr admits local stationary vacuum deformations with an axial obstruction

Let $g$ be rotating Kerr with $0<a<m$, stationary field $T$ and axial field $Z$. Write $E=I^{-}(M_{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I^{+}(M_{\mathrm{e}\mathrm{n}\mathrm{d}})$ and $H^{-}=\mathrm{\partial }{I}^{+}(M_{\mathrm{e}\mathrm{n}\mathrm{d}})$ using the conventions of Causal exterior, chronological boundaries and Cauchy data. For any open $U_0$ meeting $H^{-}\cap \bar{E}$, the Ionescu–Klainerman construction supplies a smaller open neighborhood $U\subset U_0$, diffeomorphic to the open four-ball and meeting $H^{-}$, and a smooth Lorentzian metric $\tilde{g}$ with

$\mathrm{Ric}(\tilde{g})=0,{\mathcal{L}}_T\tilde{g}=0,\tilde{g}=g\text{on }U\setminus E\mathrm{.}$

The Kerr axial field from the matching side does not extend to a Killing field on all of $U$ commuting with $T$. In the local collar construction the matching interior open side $O$ is connected and nonempty, and this is nonextendibility of the prescribed axial field there. Thus $(U,O,g,\tilde{g},T,Z)$ gives the obstructed relative datum of Relative stationary deformation and extension obstruction. Its regular off-axis Kerr coordinates are those of Analytic Kerr coordinates across the chosen nondegenerate horizon. This is a local vacuum deformation, not a globally regular asymptotically flat stationary black-hole counterexample.

Theorem 1.3 supplies the smooth vacuum metric, retained stationary field and prohibited commuting axial extension. The construction in §3 takes a regular horizon collar off the axis and bifurcation sphere; §3.2 identifies the matching interior side, and the proof starting on p. 23 carries the prescribed axial data from that side into its contradiction. The straightened horizon coordinates on p. 26 permit the matching collar to be taken connected. The existence and nonextension inputs are imported, not locally reconstructed. The source quantifies over a neighborhood U0 and produces U; it does not claim that the resulting metric fails to be Kerr in every subneighborhood of the original point.