Uniform nonlinear stability as $a \to M^-$

Summary

In asymptotically flat vacuum Einstein gravity, nonlinear stability of Kerr is currently proved with quantitative constants that deteriorate as the subextremality margin $\varepsilon = 1 - |a|/M \downarrow 0$. This problem asks for a **uniform-in-$\varepsilon$** stability theorem in a precise topology on Cauchy data: either (i) existence of a quantitative stability window whose size and decay/Morawetz constants admit bounds independent of $\varepsilon$ on compact $\varepsilon$-subintervals of $(0,1)$, or (ii) a sharp theorem identifying the optimal blow-up rate of those constants as $\varepsilon \to 0^+$. “Stability” is meant in the same Cauchy-formulation sense as the small-$|a|/M$ nonlinear theorems (convergence to a nearby Kerr solution modulo gauge and parameter modulation), not a separate extremal-black-hole asymptotic regime.

Why this matters

Near-extremal Kerr is the transition region between the standard subextremal theory and the qualitatively different extremal regime.

Exact scope

Background / setting

Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.

Equation type

PDE level: full-einstein.

Linearity

Includes or emphasizes nonlinear dynamics.

Regularity

Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.

Parameter regime

Subextremal Kerr with $0 < 1-|a|/M \le \varepsilon_0$; uniformity is in $\varepsilon$ on compact subintervals of $(0,1)$, not passage to the extremal limit $a \to M^-$.

Asymptotics

asymptotically-flat

Gauge / formulation

State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).

Status explanation

**Open (nonlinear, $\Lambda=0$ vacuum):** small-$|a|/M$ nonlinear stability is theorem-level; uniform stability as the subextremality margin tends to zero is not recorded here as proved. Linearized estimates on fixed Kerr still degenerate with surface gravity, so uniformity is a genuine analytic question—not a restatement of the unrestricted subextremal conjecture alone.

Problem statement

Fix a Cauchy formulation and data topology compatible with the nonlinear Kerr stability program (e.g. weighted Sobolev norms on a spacelike slice). Prove or disprove: there exist $\varepsilon_0>0$, $C>0$, and a modulus $\omega(\delta)$ such that for all subextremal Kerr parameters with $0 < \varepsilon = 1-|a|/M \le \varepsilon_0$, initial data $\delta$-close to the corresponding Kerr slice evolve globally to a final Kerr state **with all bootstrap constants in the stability proof bounded by $C \omega(\delta)$ uniformly in $\varepsilon$** on each fixed compact subinterval $[\varepsilon_{\min},\varepsilon_0]$. Alternatively, exhibit an obstruction (e.g. sharp blow-up of a Morawetz/trapping constant) as $\varepsilon \to 0^+$ within that program.

What is already known

• Nonlinear stability of vacuum Kerr is proved for sufficiently small $|a|/M$ (Klainerman–Szeftel).
  Regime: Nonlinear Einstein vacuum, asymptotically flat, small angular momentum per unit mass.
  Strongest unconditional nonlinear theorem toward the full subextremal conjecture.
• Linearized Teukolsky/wave decay and mode stability on fixed subextremal Kerr are developed in depth (microlocal and physical-space methods).
  Regime: Linearized gravity and scalar waves on exact Kerr.
  Standard toolbox; not equivalent to nonlinear stability for all parameters.
• Sharp Price-law exponents and nonlinear tail matching are understood in restricted settings (e.g. linearized models, Schwarzschild); sharp nonlinear Kerr curvature tails are not settled.
  Regime: Late-time asymptotics; mixed linear vs nonlinear literature.
  Locates what “sharp Price law” demands beyond integrated decay.

Progress summary: Nonlinear stability is available for small $|a|/M$; linearized decay tools on fixed Kerr exist but with $\varepsilon$-dependent constants. A uniform-in-$\varepsilon$ nonlinear theorem is the gap.

What remains open

Separate uniform-in-$\varepsilon$ control of redshift, trapping/Morawetz, and Teukolsky-type estimates from parameter-dependent losses that may be removable by sharper analysis; connect any obstruction to geometric mechanisms (near-extremal photon sphere/trapping).

Mathematical prerequisites

Degenerate redshift analysis; near-horizon scaling; semiclassical control near the photon region; uniform weighted energies in the surface-gravity parameter $\kappa$.

Completion criteria

A theorem or sharp counterexample that specifies which constants blow up (if any) as $\varepsilon \to 0^+$ and whether blow-up is intrinsic to trapping/redshift or an artifact of the chosen stability topology.

Implications if solved

Would bridge ordinary Kerr stability to extremal dynamics and sharpen what actually fails as $\kappa \to 0$.

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 Nonlinear stability of Kerr for small angular momentum (program) — Klainerman, Szeftel (2021)
  Theorem-level nonlinear stability of vacuum Kerr in a small $|a|/M$ regime; benchmark for full subextremal conjectures.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-679 — Prove stability/instability of higher-dimensional Kerr (Myers-Perry) in PDE sense for small angular momentum.
• K-630 — Prove nonlinear stability of Schwarzschild without codimension restrictions (full moduli convergence to Kerr).
• K-650 — Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
• K-204 — Uniform estimates in the surface-gravity limit $\kappa \to 0$
• K-207 — Extremal tail asymptotics versus conserved charges
• K-209 — Codimension and modulation theory for extremal endstates
• K-001 — Full nonlinear stability of subextremal Kerr
• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions