  In this paper, we study some intrinsic characterization of conformally
compact manifolds. We show that, if a complete Riemannian manifold admits an
essential set and its curvature tends to -1 at infinity in certain rate, then
it is conformally compactifiable and the compactified metrics can enjoy some
regularity at infinity. As consequences we prove some rigidity theorems for
complete manifolds whose curvature tends to the hyperbolic one in a rate
greater than 2.
