Existence and Compactness of conformally compact Einstein manifolds

Given a manifold (Mn;[h]), when is it the boundary of a conformally compact Einstein manifold (X^{n+1}; g+) with r^2g+ |_M = h for some defining function r on X^{n+1}? This problem of finding ”conformal filling in” is motivated by problems in the AdS/CFT correspondence in quantum gravity (proposed by Maldacena in 1998) and from the geometric considerations to study the structure of non-compact asymptotically hyperbolic Einstein manifolds.

In this talk, we will mainly discuss the compactness problem. That is, given a sequence of conformally compact Einstein manifolds with boundary, we will study the compactness of the sequence under assumption of the compactness of their restrictions on the boundary. I will first briefly survey some known results then report recent joint works in progress with Yuxin Ge. As applications, we will address the issue of ”uniqueness” and ”existence” results of the conformal filling in for some classes of manifolds.