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 $$ with conformal boundary $$. We establish a correspondence, near $$, between such spacetimes and their conformal boundary data on $$. More specifically, given a domain $$, we prove that the coefficients $$ and $$ (the undetermined term, or stress energy tensor) in a Fefferman-Graham expansion of the metric g from the boundary uniquely determine g near $$, provided $$ satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on $$, first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in $$ near $$, and with the pseudoconvexity degenerating in the limit at $$. As a corollary of this result, we deduce that conformal symmetries of $$ on domains $$ satisfying the GNCC extend to spacetime symmetries near $$. 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.