Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.

Summary

Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.

Why this matters

Clarifies what remains once mode stability is established.

Exact scope

Background / setting

de sitter general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: full-einstein (nonlinear stability target); linearized input only as explicit hypothesis.

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–de Sitter family (fixed $\Lambda>0$), $|a|<M$ in the usual sense for the black-hole sector; stability smallness measured on Cauchy data in the chosen nonlinear topology **after** spectral/mode-stability hypotheses are imposed.

Asymptotics

de sitter

Gauge / formulation

Conditional results depend on explicit spectral/mode-stability or gauge hypotheses; see status_explanation when curated.

Status explanation

**Conditional (explicit hypothesis):** nonlinear Kerr–de Sitter stability is asserted only after assuming a quantitative mode-stability / spectral-gap condition (no growing gauge-invariant modes) on the fixed background family; the theorem target is not unconditional over the full subextremal moduli without that spectral input.

Problem statement

Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.

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: Clarifies what remains once mode stability is established.

What remains open

Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.

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-608 — Prove unconditional nonlinear stability of slowly rotating Kerr-de Sitter.
• K-665 — Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.
• K-667 — Prove stability/instability of the Cauchy horizon for Kerr-de Sitter under linearized gravity with sharp norms.
• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
• 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).

Editorial / maintainer notes

Source manifest: N-009 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.