Prove rigorous existence and nonlinear stability/instability classification of Kerr-AdS endstates (black resonators/geons).

Summary

Prove rigorous existence and nonlinear stability/instability classification of Kerr-AdS endstates (black resonators/geons).

Why this matters

Kerr-AdS is a key superradiant laboratory.

Exact scope

Background / setting

anti de sitter general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: linearized-gravity.

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 (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.

Asymptotics

anti de sitter

Gauge / formulation

State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).

Status explanation

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Prove rigorous existence and nonlinear stability/instability classification of Kerr-AdS endstates (black resonators/geons).

What is already known

• Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.
  Regime: Linear waves; fixed background.
  Standard input for QNM expansions and superradiance discussions.
• Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).
  Regime: Nonlinear vacuum, restricted parameters.
  Closest nonlinear analogue for exterior stability conjectures.
• Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.
  Regime: Spectral theory on Kerr.
  Distinguishes partial mode stability from full expansion/completeness.

Progress summary: Context: Kerr-AdS is a key superradiant laboratory.

What remains open

Prove rigorous existence and nonlinear stability/instability classification of Kerr-AdS endstates (black resonators/geons).

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

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

• K-638 — Prove nonlinear instability of Kerr-AdS in a theorem (beyond numerics/backreaction studies).
• K-010 — Nonlinear superradiant endstates in Kerr–AdS
• K-603 — Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).
• K-684 — Prove that the linearized Einstein operator on Kerr has no embedded eigenvalues/resonances on the real axis beyond gauge.
• K-688 — Prove a fully rigorous PDE theorem for nonlinear superradiant instability of Kerr in a confining setting (mirror/AdS).
• K-404 — Zero-frequency structure and tail universality
• K-504 — Quantitative stability of the photon region and spherical null geodesics under near-Kerr perturbations
• K-505 — Threshold phenomena in separated Teukolsky-type ODEs on Kerr (zero modes, algebraically special limits, superradiant edges)

Editorial / maintainer notes

Source manifest: N-037 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.