Regularity and rigidity of asymptotically hyperbolic manifolds

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 −1−1 at infinity in certain rate, then it is conformally compactifiable and the compactified metrics can enjoy some regularity at infinity. As a consequence we prove some rigidity theorems for complete manifolds whose curvature tends to the hyperbolic one in a rate greater than 22.