Boundary Regularity for Conformally Compact Einstein Metrics in Even Dimensions

Abstract

We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Anderson (2003 Anderson , M. T. ( 2003 ). Boundary regularity, uniqueness and non-uniqueness for AH Einstein metrics on 4-manifolds . Adv. Math. 179 : 205 – 249 .[Crossref] , [Google Scholar], 2006 Anderson , M. T. ( 2006 ). Some results on the structure of conformally compact Einstein metrics. arXiv:math.DG/0402198 v4 . [Google Scholar]). Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This, together with boundary conditions that the compactification is shown to satisfy provide enough information to apply classical boundary regularity results. These results then provide local and global versions of finite boundary regularity for the components of the compactification.