This archive tracks 46 open problems in the mathematical analysis of Kerr black holes. Each problem is framed theorem-first: setup, equations, natural analytic framework, what a complete proof must establish, and what follows from solving it.

Note on precision: These are open problems — some entries are programmatic formulations rather than a single canonical theorem statement. The exact choice of gauge, asymptotic normalization, and weighted function spaces may vary across approaches.

📌 Flagship Problems

• K-001 — Full nonlinear stability of subextremal Kerr
• K-101 — Strong Cosmic Censorship threshold for Kerr interiors
• K-201 — Nonlinear codimension-1 stability of extremal Kerr with horizon hair

📐 Standard Background Notation

• Kerr parameters: mass $M > 0$, rotation $a$; subextremal $|a| < M$, extremal $|a| = M$
• Vacuum Einstein equation: $\operatorname{Ric}(g) = 0$
• Constraint equations on initial data $(Sigma, gamma, k)$:
  • $R_\gamma - |k|\gamma^2 + (\operatorname{tr}\gamma k)^2 = 0$
  • $\operatorname{div}\gamma\bigl(k - (\operatorname{tr}\gamma k)\gamma\bigr) = 0$
• Weighted norms: $H^N_delta times H^{N-1}_{delta-1}$, with $\delta < 0$ encoding asymptotic flatness
• Linear fields: scalar wave $Box_g psi = 0$, spin-$s$ Teukolsky $\mathcal{T}_s[\Psi_s] = 0$
• Interior regularity classes: $C^0$, $C^{0,1}$, $W^{1,2}_{mathrm{loc}}$, $C^2$

🌌 Cluster A — Exterior Stability (K-001–K-014)

Shared framework: Asymptotically flat vacuum data $(\Sigma,\gamma,k)$ satisfying the constraint equations and close in $H^N_\delta \times H^{N-1}_{\delta-1}$ to a Kerr or Kerr–Newman slice. Gauges: generalized harmonic, double-null, Bondi near $mathcal{I}^+$, or gauge-invariant Teukolsky. Core tools: vector-field method, $r^p$-weighted energy hierarchies, redshift and trapping analysis, parameter modulation, nonlinear bootstrap and continuity arguments.

• K-001 — Full nonlinear stability of subextremal Kerr · Foundational
• K-002 — Uniform nonlinear stability as a→M⁻ · Breakthrough