Nonlinear superradiant endstates in Kerr–AdS

Summary

Nonlinear superradiant endstates in Kerr–AdS

Why this matters

Kerr–AdS is the canonical setting where rotation, confinement, and superradiance collide.

Exact scope

Background / setting

Anti-de Sitter asymptotics as tagged.

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

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

Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.

Problem statement

Determine the theorem-level nonlinear endstate for small perturbations of Kerr–AdS in the presence of reflecting infinity and superradiance.

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 prove either instability or convergence to a precisely identified stationary or time-periodic family under explicit boundary conditions.

Mathematical prerequisites

AdS boundary value problems; long-time nonlinear dynamics; spectral instability; stable trapping; matched asymptotics.

Scope / taxonomy note

Completion criteria

A complete answer must prove either instability or convergence to a precisely identified stationary or time-periodic family under explicit boundary conditions.

Implications if solved

Would clarify one of the most dramatic known departures from the asymptotically flat Kerr story.

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 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-637 — Prove rigorous existence and nonlinear stability/instability classification of Kerr-AdS endstates (black resonators/geons).
• K-001 — Full nonlinear stability of subextremal Kerr
• K-002 — Uniform nonlinear stability as $a \to M^-$
• K-004 — Peeling and polyhomogeneous expansions at null infinity for nonlinear near-Kerr evolutions
• K-008 — Full asymptotically flat stability of the subextremal Kerr–Newman family
• K-012 — Low-regularity Kerr stability threshold
• K-013 — Formation plus relaxation to Kerr

Editorial / maintainer notes

Partial: substantial adjacent results or special cases exist, but the statement as written is not fully settled. : replace with a precise description of what is proved vs. conjectured.