Global bifurcation-sphere completion for perturbed Kerr-type interiors (vacuum, $\Lambda=0$, Dafermos–Luk class)
Summary
In asymptotically flat vacuum general relativity, the Dafermos–Luk program establishes $C^0$-stability-type statements for Cauchy horizons in dynamical black holes approaching Kerr. This problem isolates the **global interior boundary completion** step: extend that control through the bifurcation sphere and along the full future boundary of the perturbed interior (including articulation of the bifurcate null structure) in a stated regularity class, so that “interior stability” is not only local near a horizon patch. It is a geometric/PDE target distinct from exterior decay and from $\Lambda>0$ scalar QNM scans; success would sharpen what “generic” means for vacuum SCC-type questions in the rotating case.
Why this matters
Bifurcate geometry organizes the maximal extension; without a global statement, interior SCC heuristics can conflate local horizon estimates with global genericity claims. Broader “interior series” and charged/$\Lambda>0$ extensions belong in separate problem entries once this vacuum, $\Lambda=0$ core target is sharpened.
Exact scope
- Background / setting
- Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise. Cauchy horizon / interior regularity formulations of strong cosmic censorship are in play; distinguish $C^0$ vs higher regularity notions.
- Equation type
- PDE level: full-einstein.
- Linearity
- Includes or emphasizes nonlinear dynamics.
- Regularity
- State intended regularity class for extensions across horizons explicitly (e.g. $C^0$, Lipschitz, $C^2$).
- Parameter regime
- Subextremal Kerr interior up to Cauchy horizons; specify SCC regularity class ($C^0$, Lipschitz, $C^k$) in the theorem.
- 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
**Open / partial:** $C^0$ Cauchy-horizon stability in dynamical vacuum black holes is theorem-level in the Dafermos–Luk framework; extension to the **full bifurcation-sphere-inclusive** global boundary description as formulated here is not recorded as a single closed theorem on this site.
Problem statement
For vacuum Einstein evolutions in the dynamical-class considered by Dafermos–Luk (small perturbations of Kerr-like data), prove a **global** geometric statement that propagates the established Cauchy-horizon stability/control up to and including the bifurcation sphere, specifying the chart class (double-null, generalized harmonic, etc.) and the intended regularity ($C^0$, Lipschitz, or higher). The theorem target is a checkable statement about the full future interior boundary—not a broader “complete the entire interior program” umbrella.
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: Dafermos–Luk supply the primary $C^0$ interior baseline; the gap is packaging a global bifurcation-sphere-inclusive boundary theorem in a fixed gauge/regularity class.
What remains open
A complete answer must classify all future interior boundary pieces and prove regularity statements up to the bifurcation region.
Mathematical prerequisites
Double-null interior geometry; regularity propagation to bifurcate sets; characteristic PDE methods.
Completion criteria
A complete answer must classify all future interior boundary pieces and prove regularity statements up to the bifurcation region.
Implications if solved
Would finish the geometric part of the vacuum rotating interior program.
Formal verification suitability
FV: medium
Some subquestions may formalize before the full statement.
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.