In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (AdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate a correspondence statement as a unique continuation problem for PDEs: Is an asymptotically AdS solution of the Einstein equations uniquely determined by its data on its conformal boundary at infinity?
In this presentation, we establish a key step in toward a positive result; we prove an analogous unique continuation result for linear and nonlinear wave equations on fixed asymptotically AdS spacetimes satisfying a positivity condition at infinity. We show, roughly, that if a wave ϕ on this spacetime vanishes on a sufficiently large but finite portion of its conformal boundary, then ϕ must also vanish in a neighbourhood of the boundary. In particular, we highlight the analytic and geometric features of AdS spacetimes which enable this uniqueness result, as well as obstacles preventing such a result from holding in other cases.
This is joint work with Gustav Holzegel.