Boundary Regularity for Conformally Compact Einstein Metrics in Even Dimensions

TL;DR

This paper investigates boundary regularity of conformally compact Einstein metrics in even dimensions by analyzing the Ambient Obstruction tensor and applying classical boundary regularity results to establish finite regularity of metric components.

Contribution

It generalizes Anderson’s ideas to even dimensions, linking the vanishing of the Ambient Obstruction tensor to boundary regularity of Einstein metrics.

Findings

Established local boundary regularity results.

Proved global boundary regularity under certain conditions.

Connected the Ambient Obstruction tensor to boundary smoothness.

Abstract

We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. 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.