Null geodesics and improved unique continuation for waves in asymptotically anti-de Sitter spacetimes

Abstract

We consider the question of whether solutions of Klein–Gordon equations on asymptotically anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given in [, ], under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein–Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates—and hence new unique continuation results—for Klein–Gordon equations on a larger class of spacetimes than in [, ], in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful—both presently and more generally beyond this article—for treating tensorial objects with asymptotic limits at the conformal boundary.