Prove nonlinear stability of Schwarzschild without codimension restrictions (full moduli convergence to Kerr).
Summary
Nonlinear stability of Schwarzschild is proved in settings that may impose **codimension** restrictions on allowed perturbations (e.g. angular-mode restrictions). This problem asks whether those restrictions are **removable**: prove stability for general admissible asymptotically flat data near Schwarzschild, with **full moduli convergence** to a nearby Kerr family member when angular momentum is dynamically generated, or prove an obstruction if hidden instabilities appear once all modes are activated.
Why this matters
Connects Schwarzschild stability to rotation generation and Kerr endstates.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: full-einstein (nonlinear stability / moduli dynamics).
- 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
- 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
Partial results exist in adjacent regimes (see references); sharp alignment with this page’s exact target remains open.
Problem statement
Remove technical codimension restrictions in the known Schwarzschild nonlinear stability theorems—either prove stability for a class of perturbations dense in a natural AF topology, or exhibit failure—and clarify how rotation emerges in the stable regime (convergence to Kerr parameters rather than confinement to $a=0$).
What is already known
- Named papers in the reference list establish partial or neighboring results under explicit hypotheses; treat those as the proved baseline.Regime: As stated in cited references (often restricted parameters or linearized settings).Orients readers to literature without equating it with the full title-length target.
Progress summary: Manifest rationale: Connects Schwarzschild stability to rotation generation and Kerr endstates.
What remains open
Prove nonlinear stability of Schwarzschild without codimension restrictions (full moduli convergence to 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 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-650 — Formalize: equivalence of Teukolsky and Regge-Wheeler transformations in Schwarzschild and slowly rotating Kerr.
- K-002 — Uniform nonlinear stability as $a \to M^-$
- K-679 — Prove stability/instability of higher-dimensional Kerr (Myers-Perry) in PDE sense for small angular momentum.
- K-204 — Uniform estimates in the surface-gravity limit $\kappa \to 0$
- K-207 — Extremal tail asymptotics versus conserved charges
- K-209 — Codimension and modulation theory for extremal endstates
- 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.
Editorial / maintainer notes
Source manifest: N-030 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.