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 [15,16], 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 [15,16], 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.

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □gϕ+σϕ = (ϕ,∂ϕ) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to accomplish this with only very weak control on these geometries. Several of these estimates were proved in [14, 16], but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In [2], we will apply these estimates in order to consider a variant of the problem in [14], that of a truncated null cone in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used in [1] to study and to control the Bondi mass and the angular momentum under minimal conditions. 2010 Mathematics Subject Classification. 35R01 (Primary) 58J99, 53C21, 35Q75 (Secondary).

Abstract:We consider the unique continuation properties of asymptotically anti-de Sitter spacetimes by studying Klein-Gordon-type equations g φ + σ φ = G(φ, ∂φ), σ ∈ R, on a large class of such spacetimes. Our main result establishes that if φ vanishes to sufficiently high order (depending on σ ) on a sufficiently long time interval along the conformal boundary I, then the solution necessarily vanishes in a neighborhood of I. In particular, in the σ -range where Dirichlet and Neumann conditions are possible on I for the forward problem, we prove uniqueness if both these conditions are imposed. The length of the time interval can be related to the refocusing time of null geodesics on these backgrounds and is expected to be sharp. Some global applications as well as a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.

Abstract. We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. We find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry. In particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than the ones in the Minkowski space-time. The results are nearly optimal in many respects. They can be considered analogues of uniqueness from infinity results for second order elliptic operators. This work is partly motivated by questions in general relativity.