Given a manifold $(M^n, h)$, when is it the boundary of a conformally compact Einstein manifold $ (X^{n+1}, g) $, in the sense that there exists some defining function $r$ on X so that $ r^2 \, g$ is compact on the closure of X and $r^2 \, g $ restricted to M is the given metric $ h $? The model example is the n-sphere as the conformal infinity of the hyperbolic (n+1) ball..

In the special case when $n=3$, one can formulate the problem as an Dirichlet to Neumann type inverse problem. In the talk, I will report on some progress made with Yuxin Ge on the issues of the "compactness", and as an application, the "existence" and "uniqueness' of the fill in problem for a class of metrics of positive scalar curvature defined on the 3-sphere.