Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes

Abstract

We consider the unique continuation properties of asymptotically Anti-de Sitter spacetimes by studying Klein-Gordon-type equations ${\square }_g\varphi +\sigma \varphi =\mathcal{G}(\varphi ,\partial \varphi )$, $\sigma \in \mathbb{R}$, on a large class of such spacetimes. Our main result establishes that if $\varphi$ vanishes to sufficiently high order (depending on $\sigma$) on a sufficiently long time interval along the conformal boundary $\mathcal{I}$, then the solution necessarily vanishes in a neighborhood of $\mathcal{I}$. In particular, in the $\sigma$-range where Dirichlet and Neumann conditions are possible on $\mathcal{I}$ 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 a uniqueness result for gravitational perturbations are also discussed. The proof is based on novel Carleman estimates established in this setting.

50 pages, minor edits from referee comments

References in corpus (4)

• On the uniqueness and global dynamics of AdS spacetimes
• Unique continuation from infinity for linear waves
• Global uniqueness theorems for linear and nonlinear waves
• Unique continuation from infinity in asymptotically Anti-de Sitter spacetimes

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• A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes
• The bulk-boundary correspondence for the Einstein equations in asymptotically Anti-de Sitter spacetimes
• A gauge-invariant unique continuation criterion for waves in asymptotically Anti-de Sitter spacetimes
• The Near-Boundary Geometry of Einstein-Vacuum Asymptotically Anti-de Sitter Spacetimes