Arick Shao (QMUL)
Uniqueness Theorems on Asymptotically Anti-de Sitter Spacetimes
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 \phi on this spacetime vanishes
on a sufficiently large but finite portion of its conformal boundary,
then \phi 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.