The nonlinear stability conjecture for subextremal Kerr black holes

Let vacuum Cauchy data be sufficiently near data corresponding to a subextremal Kerr metric $g_{a_0,M_0}$ with $|a_0|<M_0$, and let $(\mathcal{M},g)$ be the associated maximal vacuum Cauchy development. Write $\mathcal{I}^+$ for future null infinity, $J^-(\mathcal{I}^+)$ for its causal past, and $\mathcal{H}^+$ for the event horizon. Nonlinear stability of Kerr. The spacetime $(\mathcal{M},g)$ should possess a complete null infinity whose past is bounded in the future by a smooth affine-complete event horizon, remain globally close to $g_{a_0,M_0}$ in $J^-(\mathcal{I}^+)$, and asymptotically settle down there to a nearby subextremal Kerr metric $g_{a,M}$ with $a\approx a_0$ and $M\approx M_0$. This is a fundamental open problem in classical general relativity; the paper constructs a large class of dynamical vacuum black holes with the asserted asymptotic behavior, but not the full Cauchy stability result.