Killing initial data on the timelike conformal boundary on anti-de Sitter-like spacetimes

By making use of the conformal Einstein field equations, we analyse initial-boundary value problems on anti de Sitter-like spacetimes ensuring the existence of Killing vectors via a homogeneous system of wave equations. In this problem, both the conformal boundary and the initial hypersurface are analysed with the help of the confromal constraint equations via a 3+1 formalism; this leads to an analogous problem on the boundary. Adopting a particular gauge on the conformal boundary, an obstruction to the existence of a Killing vector is found. The boundary data is made consistent with those at the initial hypersurface by means of appropriate corner conditions.