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

Gap phenomena and curvature estimates for Conformally Compact Einstein Manifolds