Compactness of asymptotically hyperbolic Einstein manifolds in dimension 4 and applications

Given a closed riemannian manifold of dimension 3 $(M^3,[h])$, when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 $(X^4,g_+)$ such that $r^2g_+|_M = h$ on the boundary $M = \partial X$ for some defining function $r$ on $X^4$? This problem is motivated by the correspondence AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds. In this talk, I discuss the compactness issue of asymptotically hyperbolic Einstein manifolds in dimension 4, that is, how the compactness on conformal infinity leads to the compactness of the compactification of such manifolds under the suitable conditions on the topology and on some conformal invariants. As an application I discuss the uniqueness problem and non-existence result. It is based on the works with Alice Chang.