Uniform estimates in the surface-gravity limit $\kappa \to 0$

Summary

Uniform estimates in the surface-gravity limit $\kappa \to 0$

Why this matters

Every near-extremal problem ultimately runs into constants that blow up as $\kappa \to 0$.

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

Develop estimates for wave and gravitational fields that remain meaningful as the subextremal surface gravity tends to zero.

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 state explicit $\kappa$-dependence in the coercive and decay estimates and identify the limiting operator or geometry.

Mathematical prerequisites

Uniform microlocal analysis; two-scale near-horizon asymptotics; replacement for standard redshift positivity.

Completion criteria

A complete answer must state explicit $\kappa$-dependence in the coercive and decay estimates and identify the limiting operator or geometry.

Implications if solved

Would unify subextremal and extremal analysis instead of treating them as disconnected cases.

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.