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

Abstract

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