Prove unconditional linear stability of Kerr (full subextremal range) in a fixed gauge, with full decay rates.

Summary

Prove a definitive **linearized Einstein vacuum** stability statement for the full subextremal Kerr family $|a|<M$ in an explicitly fixed gauge/regularity class, including sharp or integrable decay rates comparable to the Teukolsky program. Boundedness and decay for the Teukolsky master equations on fixed Kerr are now developed through the full subextremal range in the Dafermos–Holzegel–Rodnianski–Teixeira da Costa program; the remaining packaging issue is how far this implies a single global theorem for the full linearized metric perturbation in a chosen gauge without hidden small-$|a|/M$ restrictions.

Why this matters

Consolidates linear theory into a canonical reference and enables downstream nonlinear work.

Exact scope

Background / setting

asymptotically flat general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: linearized-gravity.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

Parameter regime

Fixed subextremal Kerr background with $|a|<M$; linearized perturbations off that exact solution, all parameters in the physical subextremal range.

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:** Teukolsky master equations on fixed subextremal Kerr now have global boundedness/decay theorems in the full $|a|<M$ regime (DHR–Teixeira da Costa program). What remains **open on this page’s phrasing** is a single clean **metric-level linearized Einstein** statement in a fixed gauge, with rates transparently inherited from that program—or a proved equivalence package spelling the norms.

Problem statement

Establish boundedness and quantitative decay for solutions to the **linearized Einstein vacuum equations** around subextremal Kerr, for all $|a|<M$, in a named gauge (double-null, generalized harmonic, etc.) and Sobolev class—**or** prove an equivalent formulation (e.g. master-variable/control norms) that is clearly equivalent to linearized metric stability and records the sharp rates. The Teukolsky-equation results on the Kerr exterior should be cited as the current theorem-level backbone.

What is already known

• Dafermos–Holzegel–Rodnianski–Teixeira da Costa develop **boundedness and polynomial decay** for Teukolsky equations of spins $\pm1,\pm2$ on Kerr in the **full subextremal** parameter range (frequency-based series starting arXiv:2007.07211), building on the small-$|a|/M$ case (arXiv:1711.07944).
  Regime: Linearized curvature master equations on exact subextremal Kerr, asymptotically flat exterior.
  Closest unconditional “linear stability” result at the level of gauge-invariant master variables; metric-level bookkeeping is the remaining packaging question for this entry.
• Linearized **scalar** wave decay on subextremal Kerr is classical in comparison; Teukolsky/gravitational degrees require the spin-dependent framework above.
  Regime: Scalar vs spin-2 on Kerr.
  Separates scalar templates from the gravitational master equations actually addressed by the DHR program.
• Nonlinear Kerr stability is proved only for small $|a|/M$ (Klainerman–Szeftel program); this entry concerns linearized theory but is motivated by wanting a **parameter-global** linear backbone matching the Teukolsky literature.
  Regime: Nonlinear vs linearized landscape.
  Clarifies that partial progress here is **not** claiming a new nonlinear theorem.

Progress summary: Teukolsky-level linear stability is substantially settled in the full subextremal range; metric/gauge packaging of the same information remains the active clarification target.

What remains open

Prove unconditional linear stability of Kerr (full subextremal range) in a fixed gauge, with full decay rates.

Mathematical prerequisites

Teukolsky equations; metric reconstruction from curvature components; gauge-fixed linearized Einstein; weighted energy estimates and trapping analysis on Kerr.

Completion criteria

A published-quality theorem for linearized metric perturbations or a proved equivalence to master-variable norms with explicit decay rates, for all $|a|<M$.

Implications if solved

Provides a canonical linear backbone for nonlinear bootstrap arguments without hiding small-$|a|/M$ restrictions inside linear tools.

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 Boundedness and decay for the Teukolsky equation on Kerr in the full subextremal range $|a|<M$: frequency space analysis — Dafermos, Holzegel, Rodnianski, Teixeira da Costa (2020)
  Foundational paper in the series proving Teukolsky boundedness/decay across the full subextremal Kerr range (spin $\pm1,\pm2$).
• primary Boundedness and decay for the Teukolsky equation on Kerr spacetimes I - the case $|a|\ll M$ — Dafermos, Holzegel, Rodnianski (2017)
  Earlier Teukolsky stability result for small $|a|/M$; conceptual precursor to the full-range series.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Survey context for how linearized decay tools feed (and do not automatically equal) full nonlinear Kerr stability.

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-602 — Prove nonlinear stability of Kerr for the full subextremal range |a|<M.
• K-612 — Prove generic sharp lower bounds on event-horizon flux for spin-2 Teukolsky fields.
• K-651 — Prove linear stability of Kerr in alternative gauges (radiation gauges, generalized wave gauges) with explicit gauge maps.
• K-652 — Prove nonlinear stability of Kerr under weaker asymptotic flatness (polyhomogeneous/rough null infinity assumptions).
• K-653 — Establish sharp decay for Teukolsky equation on Kerr in full range with quantitative constants usable as blackboxes.
• K-659 — Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
• K-664 — Prove robust decay estimates for wave/Teukolsky equations on perturbed Kerr backgrounds without separability.
• K-678 — Prove nonlinear stability of Kerr under polarized symmetry-breaking perturbations (intermediate symmetry classes).

Editorial / maintainer notes

Source manifest: N-001 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.