Boundary Regularity For Conformally Compact Einstein Manifolds

We mainly solve two problems in this thesis.Firstly,we deal with the boundary regularity of conformally compact Einstein metrics in dimension 4.With the Inter-mediate Schauder theory in PDE,we improve and perfect the results of Anderson and Helliwell.We show that,for a C² conformally compact Einstein metric with Cσ scalar curvature and C^1,σ mean curvature on boundary,if its boundary metric is in C^m,a（（?）m）,then the metric is Cm,α conformally compact by a C²,λ coordinates change.Secondly,we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension.We show that Cm,α conformally compact Riemannian metrics with Einstein equation vanishing to second order near boundary have conformal compactifacations that are C^m+2,α up to the boundary when Weyl curvature is in Cm,α and the boundary metric is in C^m+2,α.We also improve the regularity of defining function.Finally,we extend the Weyl Schouten theorem for Lipschitz and H² manifolds.