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.