Prove nonlinear stability of Kerr for the full subextremal range |a|<M.

Summary

In asymptotically flat vacuum Einstein gravity, prove **nonlinear** asymptotic stability of the entire subextremal Kerr family $|a|<M$: for initial data sufficiently close to any Kerr slice, the maximal development should approach a nearby Kerr solution modulo gauge and parameter modulation, with quantitative decay/Morawetz estimates **uniform** in the subextremality margin. Small-$|a|/M$ windows are theorem-level; the open gap is removing that smallness restriction.

Why this matters

Central dynamical stability conjecture.

Exact scope

Background / setting

asymptotically flat general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: full-einstein (nonlinear vacuum stability target).

Linearity

both linearized and fully nonlinear aspects

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

Parameter regime

Subextremal Kerr moduli $|a|<M$ (or stated KN/KdS extension); smallness measured in the stability topology on Cauchy data.

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

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Prove nonlinear asymptotic stability of subextremal Kerr for all $|a|<M$ under asymptotically flat vacuum Einstein dynamics, in a Cauchy formulation compatible with the nonlinear stability literature (weighted Sobolev norms on data, quantitative convergence to a final Kerr state).

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: Context: Central dynamical stability conjecture.

What remains open

Prove nonlinear stability of Kerr for the full subextremal range |a|<M.

Mathematical prerequisites

Match hypotheses to primary sources cited on this page; state minimal regularity, gauge class, and parameter windows in any claimed theorem.

Completion criteria

Prove a theorem or give a rigorous counterexample that matches the scoped statement under explicitly listed hypotheses.

Implications if solved

Impact depends on the solved formulation; sharpen once the statement is pinned to a literature-compatible theorem.

Formal verification suitability

FV: low

Global PDE or phenomenological target; lemma-level formalization may be possible after scoping.

See Formal verification for how this database uses these labels.

References

• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.
• primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016)
  Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.

Related problems

Related by shared tags

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

• K-652 — Prove nonlinear stability of Kerr under weaker asymptotic flatness (polyhomogeneous/rough null infinity assumptions).
• K-678 — Prove nonlinear stability of Kerr under polarized symmetry-breaking perturbations (intermediate symmetry classes).
• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-013 — Formation plus relaxation to Kerr
• K-601 — Prove unconditional linear stability of Kerr (full subextremal range) in a fixed gauge, with full decay rates.
• K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.

Editorial / maintainer notes

Source manifest: N-002 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.