Holographic Characterisation of Locally Anti-de Sitter Spacetimes

Abstract: It is shown that an $(n+1)$-dimensional asymptotically anti-de Sitter solution of the Einstein-vacuum equations is locally isometric to pure anti-de Sitter spacetime near the conformal boundary if and only if the boundary metric is conformally flat and (for $n \neq 4$) the boundary stress-energy tensor vanishes, subject to (i) sufficient (finite) regularity in the metric and (ii) the satisfaction of a geometric criterion on the boundary. A key tool in the proof is the Carleman estimate previously derived by the author with A. Shao, which is applied to prove a unique continuation result for the Weyl curvature at the conformal boundary given vanishing to sufficiently high order over a sufficiently long timespan.