Smooth stationary multiple-black-hole nonexistence

PDF p. 5: §1.3, last paragraph

Can there be stationary Lorentzian data $(M,g,N,T)$ with a specified asymptotically flat spacelike end, satisfying $\mathrm{Ric}(g)=0$ and $I^{+}$-regularity relative to that end, whose future event horizon has more than one connected component?

Here stationary Lorentzian data mean the smooth time-oriented Lorentzian four-manifold and Killing field of Stationary Lorentzian data and tensor conventions; the vacuum equation says that its Ricci tensor vanishes. Use the asymptotically flat end and $I^{+}$-regularity conditions of Staticity, axisymmetry and black-hole regularity hypotheses. In particular, the latter specifies the complete stationary flow, globally hyperbolic exterior and an appropriate acausal spacelike section. The future event horizon is the set ${\mathcal{E}}^{+}={\mathrm{\partial }}_{M}D\cap I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ in that definition, where $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is the flow saturation of the chosen end and $D=I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$. Multiple black holes means more than one connected component of this horizon. Neither analyticity nor a Killing-horizon generator, bifurcation surface, or nondegeneracy is assumed in the question.

Stationary Lorentzian data and tensor conventions

Staticity, axisymmetry and black-hole regularity hypotheses

The article poses this as an open question in §1.3, dated 12 November 2026. It reports exclusion in the analytic vacuum setting. As a restricted smooth partial result, Wong and Yu report at most one horizon component under their additional geometric and quantitative near–Kerr–Newman assumptions, in electrovacuum and without assumed axial symmetry. Their introduction also states that the single-horizon assumption in the perturbative theorem of Alexakis, Ionescu and Klainerman can thereby be removed. These reports do not resolve the unrestricted smooth question stated above. No later status survey or independent proof review is claimed.