Asymptotic stability conjecture for the Kerr family

Let $(\mathcal{M},g_{a,M})$ be a sub-extremal Kerr spacetime, with parameters satisfying $|a|<M$, and consider a sufficiently small perturbation of its initial data. Its maximal Cauchy development is the spacetime obtained by evolving that perturbed initial data.

Asymptotic stability of the Kerr family. The maximal Cauchy development of a small perturbation of sub-extremal Kerr initial data possesses a black hole and exterior region, and in the exterior region, the development remains close to the perturbed spacetime and asymptotically settles down to a possibly different Kerr exterior spacetime.

This is one of the fundamental open problems concerning the dynamics and stability of black holes in classical general relativity. The paper’s abstract notes that the constructed time-periodic Einstein–Klein–Gordon solutions provide an obstruction to asymptotic stability for the corresponding Klein–Gordon masses.