A proof of the instability of AdS for the Einstein--massless Vlasov system

In recent years, the AdS instability conjecture, put forward by Dafermos-Holzegel in 2006, has attracted a substantial amount of numerical and heuristic studies. Following the pioneering work of Bizon-Rostworowski, research efforts have been mainly focused on the study of the spherically symmetric Einstein-scalar field system. The first rigorous proof of the instability of AdS in the simplest spherically symmetric setting, namely for the Einstein-null dust system, was obtained in [G. Moschidis, A proof of the instability of AdS for the Einstein-null dust system with an inner mirror, 2017, arXiv:1704.08681]. In that work, the evolution was restricted to the exterior of an inner mirror placed at $r=r_{0}>0$, in order to circumvent the trivial break down of the Einstein-null dust system occurring at $r=0$; in that setting, additional considerations necessitated $r_{0}$ to shrink to $0$ with the size of the initial perturbation. In this paper, we establish the instability of AdS for the Einstein-massless Vlasov system in spherical symmetry. This will be the first proof of the AdS instability conjecture for an Einstein--matter system which is well-posed for regular initial data in the standard sense, without the addition of an inner mirror. New difficulties associated with the Einstein-massless Vlasov system (such as the need for control on the paths of non-radial geodesics in a large curvature regime) force us to develop a strategy of proof which is fundamentally different from the one employed in the case of the Einstein--null dust system, albeit still based on the interaction of beams of matter in physical space.