Compactness of conformally compact Einstein 4-manifolds II

In this paper, we establish a number of compactness results for some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier compactness results obtained by Chang-Ge [15]. In the second part of the paper, as applications, we derive some further compactness results under the perturbation condition when the L² norm of the Weyl curvature is small. We also derive the global uniqueness of conformally compact Einstein metrics on the 4-Ball constructed in the earlier work of Graham-Lee [31].