Full nonlinear stability of subextremal Kerr
Statement
Prove that asymptotically flat vacuum initial data sufficiently close to a Kerr slice with |a|<M evolve to a spacetime with complete future null infinity, a regular event horizon, and quantitative convergence to a nearby Kerr metric modulo gauge and parameter modulation.
Mathematical prerequisites
Geometric analysis of Einstein vacuum equations; null structure equations; gauge fixing; weighted Sobolev spaces; vector-field and r^p methods; trapping/superradiance control; modulation of mass and angular momentum.
Why it matters
This is the main Kerr stability conjecture in the physically relevant asymptotically flat vacuum setting.
Completion criteria
A complete answer must specify the initial-data topology, prove global exterior existence, identify the final Kerr parameters, establish quantitative decay rates in a fixed gauge, and control all nonlinear error terms for the full subextremal range.
Implications if solved
Would settle the core exterior black-hole stability problem for rotating vacuum black holes and give the exterior half of a precise cosmic censorship picture.