Prove unconditional nonlinear stability of slowly rotating Kerr-de Sitter.
Summary
Prove unconditional nonlinear stability of slowly rotating Kerr-de Sitter.
Why this matters
Important Lambda>0 testbed where spectral gaps aid decay.
Exact scope
- Background / setting
- de sitter general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: linearized-gravity.
- 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
- de sitter
- Gauge / formulation
- State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).
Status explanation
**Open / partial:** slowly rotating Kerr–de Sitter is a natural $\Lambda>0$ counterpart to the asymptotically flat Kerr program; unconditional **nonlinear** stability in a sharp topology is not recorded here as closed. Linear microlocal decay tools on Kerr–de Sitter exist, but closing a global nonlinear theorem still requires hypotheses matching the positive-cosmological-constant geometry.
Problem statement
Prove unconditional nonlinear stability of slowly rotating Kerr-de Sitter.
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: Manifest rationale: Important Lambda>0 testbed where spectral gaps aid decay.
What remains open
Prove unconditional nonlinear stability of slowly rotating Kerr-de Sitter.
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
- K-609 — Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.
- 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-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-008 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.