Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.

Summary

Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.

Why this matters

Closes conditional gaps in Lambda>0 stability.

Exact scope

Background / setting

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

Equation type

PDE level: spectral/normal-mode analysis on fixed Kerr–de Sitter (linearized sector) with nonlinear stability as the downstream 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

Kerr–de Sitter moduli in the subextremal black-hole sector; spectral bounds are stated on the **fixed** background family before any nonlinear bootstrap.

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):** quantitative mode stability is a **spectral assumption** on the fixed Kerr–de Sitter background (quasinormal-mode/gap-type bounds in relevant sectors); feeding it into nonlinear stability requires an additional bridge theorem—this entry names that split rather than asserting an unconditional nonlinear closure.

Problem statement

Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.

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: Closes conditional gaps in Lambda>0 stability.

What remains open

Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.

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-609 — Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.
• K-667 — Prove stability/instability of the Cauchy horizon for Kerr-de Sitter under linearized gravity with sharp norms.
• K-108 — Strong cosmic censorship in rotating $\Lambda>0$ black-hole interiors (conditional on a spectral–interior bridge)
• 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-606 — Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
• K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).

Editorial / maintainer notes

Source manifest: N-065 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.