Prove a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
Summary
Prove a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
Why this matters
Stationary uniqueness with Lambda>0 and charge/rotation.
Exact scope
- Background / setting
- de sitter general relativity context; see family and coupling tags for matter model.
- Equation type
- PDE level: full-einstein.
- 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 moduli $|a|<M$ (or stated KN/KdS extension); smallness measured in the stability topology on Cauchy data.
- 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 a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
What is already known
- Analytic stationary uniqueness theorems identify Kerr in the asymptotically flat vacuum class (Carter–Robinson–Mazur line).Regime: Real-analytic stationary vacuum.Classical baseline; smooth non-analytic uniqueness remains the sharp open gap for many formulations.
- Near-Kerr perturbative rigidity and Carter-type structures are studied in separability and hidden-symmetry programs.Regime: Perturbations of Kerr; operator commutators.Context for approximate operators and photon-region stability questions.
- Ernst reduction and harmonic-map formulations package stationary axisymmetric vacuum equations; sharp global uniqueness domains are formulation-dependent.Regime: 2D elliptic reductions.Explains why Ernst-domain questions must pin boundary data and function classes.
Progress summary: Context: Stationary uniqueness with Lambda>0 and charge/rotation.
What remains open
Prove a black-hole uniqueness theorem for near Kerr-Newman-de Sitter without analyticity in smooth category.
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 Hamilton–Jacobi and Schrödinger separability and integrability of the Kerr metric — Carter (1968) Fourth constant / separability structure on exact Kerr; operator-algebra backdrop for rigidity questions.
- survey Black Uniqueness Theorems — Mazur (2001) Survey of stationary uniqueness and reduction routes (including Ernst-type formulations).
Related problems
Related by shared tags
- K-308 — Rigidity and uniqueness with matter: full Kerr–Newman regime
- K-008 — Full asymptotically flat stability of the subextremal Kerr–Newman family
- K-607 — Prove nonlinear stability of Kerr-Newman in the asymptotically flat setting (full coupling).
- K-666 — Prove SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.
- K-007 — Einstein–Maxwell stability near Kerr
- K-304 — Near-Kerr rigidity with computable geometric constants
- K-305 — Kerr characterization from horizon intrinsic data
Editorial / maintainer notes
Source manifest: N-087 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.