Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
Summary
Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
Why this matters
Zero-frequency analysis is repeatedly cited as delicate.
Exact scope
- Background / setting
- asymptotically flat general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: linearized-gravity, spectral-operator.
- 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 sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
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: Zero-frequency analysis is repeatedly cited as delicate.
What remains open
Prove sharp resolvent bounds near omega=0 for Kerr linearized Einstein operator, uniform in a/M.
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 Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016) Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.
- 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-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-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-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-059 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.