Stability or obstruction for approximate Killing–Yano tensors near Kerr
Summary
If the Killing–Yano equation is small in an agreed tensor norm, must the metric be near Kerr, or can one build controlled counterexamples? Complements K-306 (Carter-type operators) with a 2-form/tensor PDE focus.
Why this matters
Killing–Yano tensors encode hidden symmetry; their stability under perturbation tests whether near-Kerr geometry inherits usable algebraic structure.
Exact scope
- Background / setting
- Four-dimensional vacuum near Kerr; local or global domains as stated.
- Equation type
- Geometric tensor PDE for $Y_{ab}$ coupled to vacuum Einstein constraints.
- Linearity
- Fully nonlinear metric perturbations; linearization may be a first step only.
- Regularity
- Elliptic systems on Riemannian slices; boundary conditions must be specified.
- Parameter regime
- Small Yano defect measured in the chosen norm vs Kerr distance functional.
- Asymptotics
- Asymptotically flat if global statements are claimed.
- Gauge / formulation
- Tensor norms must be coordinate-invariant or gauge-fixed consistently.
Status explanation
Paired with K-306: K-306 emphasizes operator commutators / Carter constants; this entry emphasizes the Killing–Yano 2-form PDE itself.
Problem statement
Consider vacuum spacetimes with a two-form $Y_{ab}$ whose Killing–Yano equation $\nabla_{(a}Y_{b)c}=0$ is small in an agreed $L^2$ or $C^0$ norm on a suitable spacelike slice or domain. Prove either (i) a quantitative stability statement toward Kerr with explicit bounds, or (ii) an explicit family of near-vacuum solutions with small Yano violation but large distance to Kerr in a stated topology.
What is already known
- Exact Killing–Yano structure and hidden symmetries on Kerr (classical).Regime: Exact Kerr geometry.Baseline “known qualitative” structure; this entry asks for **approximate** stability or obstruction.
Progress summary: Exact Yano structure on Kerr is classical; approximate versions at theorem level are not established here.
What remains open
Quantitative theorem with either stability toward Kerr or sharp counterexamples.
Mathematical prerequisites
Two-forms; Killing–Yano equations; elliptic systems on slices; perturbation theory of geometric PDE constraints.
Completion criteria
Either quantitative stability toward Kerr or explicit near-vacuum counterexamples with stated norms.
Implications if solved
Clarifies how much Kerr algebra persists under nonlinear perturbation.
Formal verification suitability
FV: medium
Tensor norms and approximate closure identities can be formalized; global nonlinear stability remains broad.
See Formal verification for how this database uses these labels.
References
- primary A spacetime characterization of the Kerr metric — Mars, M. (1999) Tensor-level rigidity context for special algebraic structures tied to Kerr.
- primary Hamilton–Jacobi and Schrödinger separability and integrability of the Kerr metric — Carter, B. (1968) Foundational hidden-symmetry / separability mechanism on Kerr related to Yano tensors.
Depends on
- K-301 — Global smooth Kerr uniqueness without analyticity
Related problems
Related by shared tags
- K-304 — Near-Kerr rigidity with computable geometric constants
- K-617 — Prove a quantitative distance-to-Kerr estimate from a small invariant (Mars-Simon-type) with computable constants.
- K-618 — Prove global Kerr uniqueness without analyticity under minimal smoothness/decay hypotheses.
- K-619 — Prove uniqueness of stationary black holes with small deviations in asymptotic charges (effective inverse problems).
- K-633 — Prove that near-Kerr spacetimes admit approximate Carter operators with controlled commutators (usable in PDE estimates).
- K-646 — Prove global uniqueness/rigidity for Kerr with minimal assumptions on horizon regularity and connectedness.