Azimuthal instabilities on extremal Kerr

We prove the existence of instabilities for the geometric linear wave equation on extremal Kerr spacetime backgrounds, which describe stationary black holes rotating at their maximally allowed angular velocity. These instabilities can be associated to non-axisymmetric azimuthal modes and are stronger than the axisymmetric instabilities discovered by Aretakis in [Are15]. The existence of non-axisymmetric instabilities follows from a derivation of very precise stability properties of solutions: we determine therefore the precise, global, leading-order, late-time behaviour of solutions supported on a bounded set of azimuthal modes via energy estimates in both physical and frequency space. In particular, we obtain sharp, uniform decay-in-time estimates and we determine the coefficients and rates of inverse-polynomial late-time tails everywhere in the exterior of extremal Kerr black holes. We also demonstrate how non-axisymmetric instabilities leave an imprint on future null infinity via the coefficients appearing in front of slowly decaying and oscillating late-time tails.