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 ( M , g ) with conformal boundary ( I , g ) . We establish a correspondence, near I , between such spacetimes and their conformal boundary data on I . More specifically, given a domain D ⊂ I , we prove that the coefficients g ( 0 ) = g and g ( n ) (the undetermined term , or stress energy tensor ) in a Fefferman-Graham expansion of the metric g from the boundary uniquely determine g near D , provided D satisfies a generalised null convexity condition (GNCC). The GNCC is a conformally invariant criterion on D , first identified by Chatzikaleas and the second author, that ensures a foliation of pseudoconvex hypersurfaces in M near D , and with the pseudoconvexity degenerating in the limit at D . As a corollary of this result, we deduce that conformal symmetries of ( g ( 0 ) , g ( n ) ) on domains D ⊂ I satisfying the GNCC extend to spacetime symmetries near D . 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.