Finite Boundary Regularity for Conformally Compact Einstein Manifolds of Dimension 4

We prove that a $4-$dimensional $C^2$ conformally compact Einstein manifold with H"older continuous scalar curvature and with $C^{m,\alpha }$ boundary metric has a $C^{m,\alpha }$ compactification. We also study the regularity of the new structure and the new defining function. This is a supplementary proof of Anderson’s work and an improvement of Helliwell’s result in dimension 4.