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

Abstract

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