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)-dimensional stationary vacuum space-times (possibly with cosmological constant Lambda is an element of R) that coincide up to order one along a timelike hypersurface J are isometric in a neighbourhood of J. We further prove that KIDS of partial derivative M extend to Killing vectors near 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 partial derivative M of conformal Killing vectors of conformal infinity which leave the undetermined Fefferman-Graham term invariant is also established.