Vanishing-$\Lambda$ black hole stability conjecture

Let ${\bf g}_{M,a}$ be a subextremal Kerr metric. Let $(\Sigma_0,g_0,k_0)$ be a suitably small perturbation of its initial data, where $\Sigma_0$ is a 3-dimensional manifold, $g_0$ is a Riemannian metric on $\Sigma_0$, and $k_0$ is a symmetric 2-tensor satisfying the constraint equations. Suppose $(\Sigma_\Lambda,g_\Lambda,k_\Lambda)$ is a family of initial data converging in an appropriate sense to $(\Sigma_0,g_0,k_0)$ and is itself a small perturbation of the initial data of the subextremal Kerr-de Sitter metric ${\bf g}_{M,a,\Lambda}$. Vanishing-$\Lambda$ black hole stability conjecture. The evolution of $(\Sigma_\Lambda,g_\Lambda,k_\Lambda)$ converges in an appropriate sense to the evolution of $(\Sigma_0,g_0,k_0)$ on regions where both $t,r\lesssim\Lambda^{-\frac12}$. The conjecture concerns the singular vanishing-$\Lambda$ limit: Kerr perturbations decay polynomially, whereas slowly rotating Kerr-de Sitter perturbations decay exponentially, and the two spacetimes have different asymptotic structures. The required convergence in the expanding regions $t,r\lesssim\Lambda^{-1/2}$ remains open.