Some Geometrical Problems On Conformally Compact Manifolds

This thesis is concerned with the study of two problems. One is the intrinsic char-acterization of conformally compact metrics; the other is the short-time existence ofstatic flows under certain initial conditions. It is interesting that how we can obtainthe regularity up to the boundary of the conformally compact metrics just under theintrinsic conditions of the asymptotically hyperbolic manifolds. This thesis shows that,if a complete Riemannian manifold admits an essential set and its curvature tends to1 at infinity in certain rate, then it is conformally compactifiable and the compactifiedmetrics can enjoy some regularity at infinity. As consequences, this thesis generalizesthe Shi-Tian rigidity result. Static flow is a kind of geometric flows related to the staticsolutions of Einstein vacuum equations, whose stationary points of the evolution equa-tions can be interpreted as a class of static solutions of Einstein vacuum equations. Thisthesis studies short-time existence of static flows on a class of complete non-compactmanifolds and also gives the asymptotic expansions of the triple of static Einstein vac-uum at conformal infinity.