Low-regularity Kerr stability threshold
Summary
Low-regularity Kerr stability threshold
Why this matters
Current high-regularity proofs are powerful but not obviously close to the true threshold.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.
- Equation type
- PDE level: full-einstein.
- Linearity
- Includes or emphasizes nonlinear dynamics.
- Regularity
- Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.
- 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
Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.
Problem statement
Find the minimal differentiability and asymptotic-flatness assumptions under which a Kerr stability theorem can hold.
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: Partial progress exists in adjacent regimes;
What remains open
A complete answer must give a stability theorem at explicit regularity $N$ and falloff $\delta$, or produce a sharp obstruction below that threshold.
Mathematical prerequisites
Low-regularity hyperbolic PDE; rough wave gauges; bilinear and null-structure estimates under weak decay; weighted Sobolev spaces.
Completion criteria
A complete answer must give a stability theorem at explicit regularity $N$ and falloff $\delta$, or produce a sharp obstruction below that threshold.
Implications if solved
Would reveal how robust the Kerr stability mechanism really is.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
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-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.
- K-632 — Quantify how much Kerrness can be certified from finite-radius curvature invariants (numerical-relativity certification).
Editorial / maintainer notes
Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.