Nonlinear stability of slowly rotating Kerr-de Sitter

Summary

The original expansion prompt asked for unconditional nonlinear stability of slowly rotating Kerr–de Sitter. That statement is solved in the literature; this page is retained as a solved literature pointer, not as an open problem.

Why this matters

Important $\Lambda>0$ testbed where spectral gaps aid decay, and a clean example of a literature-solved stability problem that should not remain in the open-problem list.

Exact scope

Background / setting

Four-dimensional Einstein vacuum equations with positive cosmological constant, $\operatorname{Ric}(g)-\Lambda g=0$.

Equation type

PDE level: full-einstein.

Linearity

Fully nonlinear Einstein evolution, with linearized stability as a proof input in Fang's route.

Regularity

Regularity and constraint hypotheses are those in the cited Hintz–Vasy and Fang theorems.

Parameter regime

Slowly rotating Kerr–de Sitter parameters with $\Lambda>0$; this closure does not cover the full subextremal Kerr–de Sitter range.

Asymptotics

de sitter

Gauge / formulation

Gauge/final-parameter modulation as in the cited nonlinear stability theorems.

Status explanation

**Solved — literature closure submitted by abdulrahimiqbal.** K-608 asks for unconditional nonlinear stability of slowly rotating Kerr–de Sitter. This is exactly the slowly rotating $\Lambda>0$ stability theorem proved by Hintz–Vasy and later reproved by Fang. This is not a new proof claim; full-subextremal Kerr–de Sitter stability, conditional mode-stability programs, and Kerr–de Sitter interior/SCC problems remain separate open targets.

Problem statement

Literature pointer: nonlinear stability of the slowly rotating Kerr–de Sitter family as solutions of the vacuum Einstein equations with positive cosmological constant.

What is already known

• Hintz and Vasy establish full global nonlinear stability of the Kerr–de Sitter family for small angular momenta, without symmetry assumptions on the initial data.
  Regime: Vacuum Einstein with $\Lambda>0$, slowly rotating Kerr–de Sitter.
  Solves the statement as originally imported into this entry.
• Fang gives a later proof of nonlinear stability for slowly rotating Kerr–de Sitter using a bootstrap argument and lower regularity assumptions than the Nash-Moser route.
  Regime: Vacuum Einstein with $\Lambda>0$, slowly rotating Kerr–de Sitter.
  Independent confirmation/alternative proof route for the solved statement.
• Fang's linear stability theorem supplies the companion gauge-fixed linear theory used in the later nonlinear proof.
  Regime: Linearized Einstein equations around slowly rotating Kerr–de Sitter.
  Separates the nonlinear closure from related full-subextremal or interior problems.

Progress summary: Solved as written: Hintz–Vasy prove slowly rotating Kerr–de Sitter nonlinear stability, and Fang gives a later nonlinear proof supported by a companion linear stability theorem.

What remains open

The slowly rotating Kerr–de Sitter statement is solved. Sharper open descendants should be split into new entries: full subextremal Kerr–de Sitter, lower regularity, explicit constants, or variant gauges/boundary formulations.

Solution pointer

Small perturbations of slowly rotating Kerr–de Sitter initial data for the Einstein vacuum equations with $\Lambda>0$ exist globally in the relevant exterior/cosmological region and converge exponentially, modulo gauge and final-parameter modulation, to a nearby Kerr–de Sitter metric.

• Attributed to: Hintz and Vasy; alternative proof by Fang (2018)
• Citation key: hintz-vasy-kds-nonlinear-stability

Mathematical prerequisites

Einstein vacuum equations with $\Lambda>0$; microlocal analysis on asymptotically de Sitter spaces; constraint damping/gauge fixing; nonlinear stability bootstraps.

Scope / taxonomy note

Completion criteria

Already met for the slowly rotating Kerr–de Sitter formulation by the references listed above.

Implications if solved

Provides the positive-cosmological-constant stability baseline; sharper descendants should be stated as separate open problems.

Formal verification suitability

FV: low

Global nonlinear PDE theorem; formalization would likely begin with scoped linear or microlocal lemmas.

See Formal verification for how this database uses these labels.

References

• primary The global non-linear stability of the Kerr-de Sitter family of black holes — Hintz, Vasy (2018)
  Establishes the nonlinear stability result for small angular momenta in Kerr–de Sitter, solving the imported statement.
• primary Nonlinear Stability of the Slowly-Rotating Kerr-de Sitter Family — Fang, Allen Juntao (2026)
  Alternative proof of nonlinear stability for the slowly rotating Kerr–de Sitter family.
• primary Linear Stability of the Slowly-Rotating Kerr-de Sitter Family — Fang, Allen Juntao (2026)
  Companion gauge-fixed linear stability theorem used in Fang's nonlinear proof route.

Related problems

Related by shared tags

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

• 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-010 — Nonlinear superradiant endstates in Kerr–AdS

Editorial / maintainer notes

Closure proposal submitted by abdulrahimiqbal. This is a literature closure: credit for the theorem belongs to Hintz–Vasy and Fang.

Source manifest: N-008 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository. Closure proposal submitted by abdulrahimiqbal as a literature closure, not an original proof.