Codimension and modulation theory for extremal endstates

Summary

Codimension and modulation theory for extremal endstates

Why this matters

The phrase 'codimension-1 extremal stability' needs a concrete theorem-level dynamical meaning.

Exact scope

Background / setting

Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.

Equation type

PDE level: full-einstein.

Linearity

Primarily stationary or linearized reductions unless the statement says otherwise.

Regularity

Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.

Parameter regime

Extremal or near-extremal Kerr-type parameters; quantify smallness of $|1-|a|/M|$ or surface gravity $κ$ in any claim.

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

Build a nonlinear stable-manifold/modulation framework showing precisely which perturbations converge to extremal versus subextremal Kerr endstates.

What is already known

• Aretakis instability and conserved charges on extremal horizons are established for scalar test fields; spin-2 and nonlinear extremal dynamics are much less complete.
  Regime: Extremal horizons; often linear scalar.
  Shows qualitative difference from subextremal decay.
• Subextremal nonlinear Kerr stability is known for small $|a|/M$; uniformity as $|a|\to M$ is not a corollary.
  Regime: Nonlinear vacuum, restricted subextremal window.
  Separates near-extremal uniformity from existing subextremal theorems.
• Near-horizon NHEK limits capture extremal mode structure but matching to global Kerr is an open PDE bridge.
  Regime: Near-horizon scaling limits.
  Clarifies what NHEK analyses do and do not imply globally.

Progress summary: Partial progress exists in adjacent regimes;

What remains open

A complete answer must define the stable manifold, prove transversality, and describe the flow off the manifold.

Mathematical prerequisites

Invariant manifold theory in PDE; modulation equations; degenerate parameter dynamics; constraint control.

Completion criteria

A complete answer must define the stable manifold, prove transversality, and describe the flow off the manifold.

Implications if solved

Would turn the current conjectural picture of extremal dynamics into a structured phase portrait.

Formal verification suitability

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

References

• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.
• 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.

Related problems

Related by shared tags

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

• K-002 — Uniform nonlinear stability as $a \to M^-$
• K-679 — Prove stability/instability of higher-dimensional Kerr (Myers-Perry) in PDE sense for small angular momentum.

Editorial / maintainer notes

Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.