Nonlinear stability of Kerr for all spins below extremal
In plain words
Proofs that a spinning black hole is stable appeared only in 2021-2022, and only for slow spin. In June and September 2026 two independent proofs were posted that cover every spin below the maximum, and these await verification.
Precise statement
Prove that the maximal development of asymptotically flat vacuum data close to subextremal $\operatorname{Kerr}(a_0, M_0), \mid a_0\mid < M_0$, has complete future null infinity and a domain of outer communication converging to a nearby $\operatorname{Kerr}(a_f, M_f)$. Residual question: confirm the 2026 proofs and close the gap between their data classes (structured data with polyhomogeneous expansions plus an O(r^-3-eps) remainder in one, structureless O(r^-3/2-delta) decay in the other).
What would settle it
Refereed acceptance of a proof for $\mid a \mid < M$ covering initial data with the physically expected decay rate.
Status in the literature
Unverified note
Hintz (arXiv:2606.28253, June 2026) and Szeftel (arXiv:2609.40068, September 2026) posted proofs for $\left|a\right| < M$; both are unrefereed preprints as of October 2026.