Prove that near-Kerr spacetimes admit a robust foliation by generalized GCM spheres with quantified control.
Summary
Prove that near-Kerr spacetimes admit a robust foliation by generalized GCM spheres with quantified control.
Why this matters
Structural geometric ingredient for stability proofs.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: full-einstein.
- 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
Partial results exist in adjacent regimes (see references); sharp alignment with this page’s exact target remains open.
Problem statement
Prove that near-Kerr spacetimes admit a robust foliation by generalized GCM spheres with quantified control.
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: Structural geometric ingredient for stability proofs.
What remains open
Prove that near-Kerr spacetimes admit a robust foliation by generalized GCM spheres with quantified control.
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-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
- K-013 — Formation plus relaxation to Kerr
- K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
- K-604 — Establish a sharp peeling/polyhomogeneity theorem at future null infinity ($\mathcal{I}^+$) for nonlinear near-Kerr evolutions.
- K-605 — Prove sharp nonlinear Price-law tails for curvature in near-Kerr vacuum.
- K-616 — Prove robust control of trapping geometry under dynamical near-Kerr perturbations in a sharp topology.
Editorial / maintainer notes
Source manifest: N-074 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.