Global smooth Kerr uniqueness without analyticity
Summary
Show any smooth stationary asymptotically flat vacuum black-hole exterior satisfying standard hypotheses is Kerr; perturbative/near-Kerr rigidity exists, but global smooth uniqueness without analyticity remains open.
Why this matters
Stationary uniqueness is the equilibrium counterpart to dynamical Kerr stability; removing analyticity matches the natural smooth category for PDE.
Exact scope
- Background / setting
- Four-dimensional Einstein vacuum, stationary Killing field, asymptotically flat ends, nondegenerate horizon (standard uniqueness hypotheses—sharpen as in references).
- Equation type
- Stationary reduction of vacuum Einstein; elliptic-hyperbolic coupled systems.
- Linearity
- Stationary (elliptic-type) problem, not a time-evolution claim.
- Regularity
- Smooth ($C^\infty$) or high Sobolev—must match the uniqueness program cited.
- Parameter regime
- Subextremal Kerr moduli $(M,a)$ as the target classification.
- Asymptotics
- Asymptotically flat with stated decay on the stationary slices.
- Gauge / formulation
- Stationary coordinates / maximal-isothermal gauges as in rigidity proofs.
Status explanation
Partial: strong classical theorems exist under analyticity or in structured perturbative regimes; the sharp global smooth statement remains open.
Problem statement
Prove that any smooth, stationary, asymptotically flat vacuum black-hole exterior with a nondegenerate connected event horizon (and standard asymptotic/axisymmetry hypotheses as in the cited rigidity literature) is isometric to a Kerr exterior—without assuming real-analyticity of the metric.
What is already known
- Classical uniqueness of Kerr among analytic stationary asymptotically flat vacuum black holes (Carter–Robinson–Mazur line; see survey).Regime: Analytic stationary vacuum.Qualitative uniqueness is classical in the analytic category; the open issue is lowering regularity.
- Partial results toward uniqueness without analyticity via unique continuation and perturbative rigidity (see e.g. Alexakis–Ionescu–Klainerman program).Regime: Smooth/near-Schwarzschild or structured perturbative regimes as in the paper.Shows nontrivial progress but does not close the global smooth statement as commonly formulated.
Progress summary: Analytic-class uniqueness is classical; removing analyticity has theorem-level partial results but no full global smooth proof in the generality often quoted informally.
What remains open
A global theorem matching the smooth, asymptotically flat, stationary vacuum black-hole exterior hypotheses without embedding in an analytic class.
Mathematical prerequisites
Rigidity theory; Carleman unique continuation; stationary reduction; horizon geometry; elliptic-hyperbolic coupled systems.
Completion criteria
State a sharp hypothesis set and prove global isometry to Kerr in the smooth category.
Implications if solved
Completes the classical uniqueness story in the physically natural smooth setting.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
See Formal verification for how this database uses these labels.
References
- primary Uniqueness of smooth stationary black holes in vacuum — Alexakis, Ionescu, Klainerman (2009) Representative partial progress on stationary uniqueness without analyticity in a structured setting.
- survey Black Uniqueness Theorems — Mazur (2001) Survey of Carter–Robinson–Mazur analytic uniqueness line and historical hypotheses.
Unlocks (other problems list this one as a dependency)
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