Finite Boundary Regularity for Conformally Compact Einstein Manifolds of Dimension 4

TL;DR

This paper proves boundary regularity results for 4-dimensional conformally compact Einstein manifolds with less smooth boundary data, improving previous results and providing supplementary proofs.

Contribution

It establishes boundary regularity for 4D conformally compact Einstein manifolds with lower regularity assumptions, extending and refining prior work.

Findings

Boundary regularity achieved under weaker smoothness conditions.

Provides supplementary proof of Anderson's results.

Improves upon Helliwell's regularity results in dimension 4.

Abstract

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.