On compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III

In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $d\ge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $d\ge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.