On uniqueness of stationary vacuum black holes

TL;DR

The paper proves the Kerr uniqueness for stationary, asymptotically flat, analytic, four-dimensional vacuum black holes with connected, non-degenerate horizons by reducing the Einstein equations to a harmonic map problem into hyperbolic space via the Ernst potential on the orbit space. It develops a comprehensive framework: topological regularity (I^+-regularity), horizon sections, area-function structures, and boundary data, then uses the orbit-space reduction to relate the problem to a two-dimensional harmonic-map equation with prescribed axis data. The authors establish the existence and uniqueness of the harmonic-map solution corresponding to Kerr, show that horizons are Killing horizons in the analytic vacuum setting, and treat both rotating and non-rotating horizon cases within a unified approach, ultimately deriving Schwarzschild/Kerr classification under the stated hypotheses. They also discuss limitations (analyticity) and potential extensions to electro-vacuum and higher-dimensional cases, outlining the next steps toward a more general uniqueness theorem.

Abstract

We prove uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes.

Figures (3)

• Figure 1.1: The hypersurface ${\mycal S}$ from the definition of $I^+$--regular ity.
• Figure 2.1: ${\mycal S}_{\mathrm{ext}}$, ${\mycal M}_{\mathrm{ext}}$, together with the future and the past of ${\mycal M}_{\mathrm{ext}}$. One has ${\mycal M}_{\mathrm{ext}}\subset I^\pm({\mycal M}_{\mathrm{ext}})$, even though this is not immediately apparent from the figure. The domain of outer communications is the intersection $I^+({\mycal M}_{\mathrm{ext}})\cap I^-({\mycal M}_{\mathrm{ext}})$, compare Figure \ref{['fregu']}.
• Figure 5.1: The quotient space $\mathcal{Q}$ and its double $\,\widehat{\!\mathcal{Q}}$.

Theorems & Definitions (12)

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