Prove SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.

Summary

Prove SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.

Why this matters

Makes Lambda>0 SCC precise and verifiable.

Exact scope

Background / setting

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

Equation type

PDE level: spectral-operator.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

Parameter regime

Subextremal Kerr interior up to Cauchy horizons; specify SCC regularity class ($C^0$, Lipschitz, $C^k$) in the theorem.

Asymptotics

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 SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.

What is already known

• $C^0$-extendibility and weak regularity across Cauchy horizons are understood in substantial $Lambda=0$ vacuum settings (Dafermos–Luk program); higher regularity and $Lambda>0$ charged models require separate hypotheses.
  Regime: Dynamical vacuum near Kerr, $Lambda=0$ baseline; contrast with $Lambda>0$ scalar scans.
  Sets what “partial” interior control means before claiming generic blow-up or extendibility.
• Linear scalar and Teukolsky-type decay on fixed subextremal Kerr exteriors is highly developed and feeds conditional interior instability heuristics.
  Regime: Linearized fields on exact Kerr/Kerr–Newman.
  Supplies quantitative decay exponents used in bridge hypotheses to inner horizons.
• Polyhomogeneous/null-infinity technology exists for nonlinear Minkowski and some linearized Kerr contexts; sharp nonlinear near-Kerr peeling is not packaged as one theorem.
  Regime: Null infinity / linearized models.
  Separates radiation asymptotics from interior SCC targets.

Progress summary: Context: Makes Lambda>0 SCC precise and verifiable.

What remains open

Prove SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.

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 The interior of dynamical vacuum black holes I: $C^0$-stability of the Kerr Cauchy horizon — Dafermos, Luk (2017)
  Foundational interior/Cauchy-horizon stability in $Lambda=0$ vacuum; context for SCC-type questions.
• primary Stability of Minkowski space and polyhomogeneity of the metric — Hintz, Vasy (2017)
  Sharp null-infinity asymptotics in a nonlinear vacuum setting; template for peeling/polyhomogeneous questions.

Related problems

Related by shared tags

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

• K-687 — Prove a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
• K-007 — Einstein–Maxwell stability near Kerr
• K-008 — Full asymptotically flat stability of the subextremal Kerr–Newman family
• K-108 — Strong cosmic censorship in rotating $\Lambda>0$ black-hole interiors (conditional on a spectral–interior bridge)
• K-308 — Rigidity and uniqueness with matter: full Kerr–Newman regime
• K-606 — Prove Kerr-Newman linear stability for a wider parameter regime beyond weak charge/slow rotation.
• K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
• K-642 — Prove analogous Cauchy-horizon instability results for coupled gravito-electromagnetic perturbations on Kerr-Newman.

Editorial / maintainer notes

Source manifest: N-066 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.