Full nonlinear stability of subextremal Kerr

Summary

Prove asymptotically flat vacuum stability for the entire subextremal Kerr family; small-angular-momentum nonlinear regimes have theorem-level results, while the unrestricted subextremal range remains open.

Why this matters

This is the central exterior vacuum stability benchmark for rotating black holes in asymptotically flat general relativity; a full theorem would anchor the nonlinear theory of Kerr as a dynamical attractor.

Exact scope

Background / setting

Four-dimensional Einstein vacuum equations, asymptotically flat Cauchy development from a spacelike slice, comparing to the subextremal Kerr family ($|a|<M$).

Equation type

Full nonlinear vacuum Einstein equations (not a fixed-background model).

Linearity

Nonlinear evolution; linearized theory is a tool, not the target claim.

Regularity

As in the cited stability program: strong enough Sobolev/decay assumptions on data to close nonlinear estimates; exact minimal regularity is part of a complete solution.

Parameter regime

Target is the full subextremal range $|a|<M$ at fixed mass scale. Current theorem-level nonlinear stability is established in restricted regimes (notably small $|a|/M$), not for all subextremal parameters simultaneously.

Asymptotics

Asymptotically flat (no cosmological constant in the primary formulation).

Gauge / formulation

Stability is understood modulo diffeomorphisms and modulation of Kerr parameters $(M,a)$ along the flow.

Status explanation

Marked “partial” because restricted-parameter nonlinear theorems exist, while the statement as written (full subextremal family without small-$|a|$ restriction) remains open. “Partial” here does not mean the unrestricted case is nearly proved—only that some nontrivial nonlinear regimes are settled.

Problem statement

Prove that asymptotically flat vacuum initial data sufficiently close to a Kerr slice with $|a|<M$ evolve to a spacetime with complete future null infinity, a regular event horizon, and quantitative convergence to a nearby Kerr metric modulo gauge and parameter modulation—without restricting to $|a|\ll M$ once a full theorem is available.

What is already known

• Nonlinear stability of Kerr for sufficiently small angular momentum per unit mass (a theorem-level result in the Klainerman–Szeftel program for $|a|/M$ small).
  Regime: Vacuum, asymptotically flat, $|a|/M$ small in the sense of that work.
  Establishes a genuine nonlinear stability theorem for Kerr in a nontrivial open subset of parameters; it does not yet cover the full subextremal family.
• Large body of linearized decay and mode stability results on fixed subextremal Kerr backgrounds.
  Regime: Linearized Einstein vacuum on exact Kerr.
  Necessary input and partial evidence; not equivalent to a nonlinear stability theorem for the full $|a|<M$ family.

Progress summary: Nonlinear Kerr stability is proved for small $|a|/M$; the full subextremal family remains a major open frontier. Linearized decay on fixed Kerr is much better developed.

What remains open

Close a global nonlinear stability theorem for asymptotically flat vacuum data near arbitrary subextremal Kerr slices (all $|a|<M$), including sharp modulation of parameters, uniform trapping/superradiance estimates, and a complete causal/geometric description of the developed spacetime.

Mathematical prerequisites

Geometric analysis of Einstein vacuum equations; null structure equations; gauge fixing; weighted Sobolev spaces; vector-field and $r^p$ methods; trapping/superradiance control; modulation of mass and angular momentum.

Completion criteria

A complete answer must specify the initial-data topology, prove global exterior existence, identify the final Kerr parameters, establish quantitative decay rates in a fixed gauge, and control nonlinear errors uniformly over the claimed subextremal parameter window.

Implications if solved

Would settle the core exterior black-hole stability problem for rotating vacuum black holes and supply the exterior half of a precise strong cosmic censorship–compatible picture in the vacuum setting.

Formal verification suitability

FV: low

Full nonlinear vacuum Einstein stability is a global PDE program; formalization should target lemmas and restricted regimes first.

See Formal verification for how this database uses these labels.

References

• primary Nonlinear stability of Kerr (small angular momentum regime) — Klainerman, Szeftel (sequence) (2021)
  Primary stability theorem in a small-$|a|/M$ window; anchors what “partial” refers to for nonlinear vacuum Kerr.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Survey of the program, hypotheses, and relation between linear tools and the nonlinear theorem.

Unlocks (other problems list this one as a dependency)

• K-402 — Nonlinear QNMs from full Einstein evolution

Related problems

Related by shared tags

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

• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.
• K-605 — Prove sharp nonlinear Price-law tails for curvature in near-Kerr vacuum.
• K-616 — Prove robust control of trapping geometry under dynamical near-Kerr perturbations in a sharp topology.
• K-632 — Quantify how much Kerrness can be certified from finite-radius curvature invariants (numerical-relativity certification).