Finite Boundary Regularity for Conformally Compact Einstein Manifolds of Dimension 4

Abstract

We prove that a $4-$dimensional $C^2$ conformally compact Einstein manifold with Hölder 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.

22 pages. Changed the proof in page 17 and 18 and removed the condition "and the mean curvature $H\in C^{1,\sigma }(\partial M)$" in Theorem 1.1. Removed and changed some sentences concerning the mean curvature in other places