Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).

Summary

Beyond stability-as-convergence-to-Kerr, build a **quantitative scattering** framework: existence of scattering operators (or wave operators) comparing nonlinear solutions with linearized radiation on Kerr, asymptotic completeness in a stated topology, and stable mapping from Cauchy data to radiation coefficients at $\mathcal{I}^+$. This upgrades decay statements into a PDE-level scattering picture comparable to linear scattering theory, but on the nonlinear near-Kerr class.

Why this matters

Upgrades decay to Kerr into a full scattering map with data-to-radiation control.

Exact scope

Background / setting

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

Equation type

PDE level: full-einstein with linearized scattering comparison.

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 (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.

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

For asymptotically flat vacuum evolutions near Kerr, prove a theorem-level scattering-type statement (e.g. existence of wave operators, asymptotic completeness, or a quantitative comparison map) between nonlinear solutions and a linearized radiation model on the final Kerr background, with norms specified in the stability topology.

What is already known

• Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.
  Regime: Linear waves; fixed background.
  Standard input for QNM expansions and superradiance discussions.
• Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).
  Regime: Nonlinear vacuum, restricted parameters.
  Closest nonlinear analogue for exterior stability conjectures.
• Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.
  Regime: Spectral theory on Kerr.
  Distinguishes partial mode stability from full expansion/completeness.

Progress summary: Context: Upgrades decay to Kerr into a full scattering map with data-to-radiation control.

What remains open

Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).

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

• 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.
• 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 problems

Related by shared tags

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

• K-660 — Establish a full nonlinear ringdown plus tail decomposition for near-Kerr vacuum spacetimes.
• K-662 — Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).
• K-683 — Prove a sharp characterization of the Kerr trapped set as a normally hyperbolic invariant manifold uniformly in a/M.
• K-684 — Prove that the linearized Einstein operator on Kerr has no embedded eigenvalues/resonances on the real axis beyond gauge.
• K-506 — High-frequency Kerr quasinormal-mode laws with explicit remainder bounds

Editorial / maintainer notes

Source manifest: N-003 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.