Prove robust decay estimates for wave/Teukolsky equations on perturbed Kerr backgrounds without separability.

Summary

Prove robust decay estimates for wave/Teukolsky equations on perturbed Kerr backgrounds without separability.

Why this matters

Key step toward nonlinear theory without hidden symmetries.

Exact scope

Background / setting

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

Equation type

PDE level: linearized-gravity.

Linearity

linearized

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 robust decay estimates for wave/Teukolsky equations on perturbed Kerr backgrounds without separability.

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: Key step toward nonlinear theory without hidden symmetries.

What remains open

Prove robust decay estimates for wave/Teukolsky equations on perturbed Kerr backgrounds without separability.

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 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 problems

Related by shared tags

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

• K-601 — Prove unconditional linear stability of Kerr (full subextremal range) in a fixed gauge, with full decay rates.
• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-612 — Prove generic sharp lower bounds on event-horizon flux for spin-2 Teukolsky fields.
• K-651 — Prove linear stability of Kerr in alternative gauges (radiation gauges, generalized wave gauges) with explicit gauge maps.
• K-652 — Prove nonlinear stability of Kerr under weaker asymptotic flatness (polyhomogeneous/rough null infinity assumptions).
• K-653 — Establish sharp decay for Teukolsky equation on Kerr in full range with quantitative constants usable as blackboxes.
• K-659 — Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
• K-678 — Prove nonlinear stability of Kerr under polarized symmetry-breaking perturbations (intermediate symmetry classes).

Editorial / maintainer notes

Source manifest: N-064 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.