The Bulk-Boundary Correspondence for the Einstein Equations in Asymptotically Anti-de Sitter Spacetimes

In this paper, we consider vacuum asymptotically anti-de Sitter spacetimes [Formula: see text] with conformal boundary [Formula: see text] . We establish a correspondence, near [Formula: see text] , between such spacetimes and their conformal boundary data on [Formula: see text] . More specifically, given a domain [Formula: see text] , we prove that the coefficients [Formula: see text] and [Formula: see text] (the undetermined term, or stress energy tensor) in a Fefferman–Graham expansion of the metric g from the boundary uniquely determine g near [Formula: see text] , provided [Formula: see text] satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on [Formula: see text] , first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in [Formula: see text] near [Formula: see text] , and with the pseudoconvexity degenerating in the limit at [Formula: see text] . As a corollary of this result, we deduce that conformal symmetries of [Formula: see text] on domains [Formula: see text] satisfying the GNCC extend to spacetime symmetries near [Formula: see text] . The proof, which does not require any analyticity assumptions, relies on three key ingredients: (1) a calculus of vertical tensor-fields developed for this setting; (2) a novel system of transport and wave equations for differences of metric and curvature quantities; and (3) recently established Carleman estimates for tensorial wave equations near the conformal boundary.