Unique continuation and extensions of Killing vectors at boundaries for stationary vacuum space-times

Generalizing Riemannian theorems of Anderson–Herzlich and Biquard, we show that two (n+1)(n+1)-dimensional stationary vacuum space-times (possibly with cosmological constant Λ∈RΛ∈R) that coincide up to order one along a timelike hypersurface TT are isometric in a neighbourhood of TT. We further prove that KIDS of ∂M∂M extend to Killing vectors near ∂M∂M. In the AdS type setting, we show unique continuation near conformal infinity if the metrics have the same conformal infinity and the same undetermined term. Extension near ∂M∂M of conformal Killing vectors of conformal infinity which leave the undetermined Fefferman–Graham term invariant is also established.