  In this paper we first use the result in $[12]$ to remove the assumption of
the $L^2$ boundedness of Weyl curvature in the gap theorem in $[9]$ and then
obtain a gap theorem for a class of conformally compact Einstein manifolds with
very large renormalized volume. We also uses the blow-up method to derive
curvature estimates for conformally compact Einstein manifolds with large
renormalized volume. The second part of this paper is on conformally compact
Einstein manifolds with conformal infinities of large Yamabe constants. Based
on the idea in $[15]$ we manage to give the complete proof of the relative
volume inequality $(1.9)$ on conformally compact Einstein manifolds. Therefore
we obtain the complete proof of the rigidity theorem for conformally compact
Einstein manifolds in general dimensions with no spin structure assumption (cf.
$[29, 15]$) as well as the new curvature pinch estimates for conformally
compact Einstein manifolds with conformal infinities of very large Yamabe
constant. We also derive the curvature estimates for conformally compact
Einstein manifolds with conformal infinities of large Yamabe constant.
