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

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