Boundary Regularity for Asymptotically Hyperbolic Metrics with Smooth Weyl Curvature

TL;DR

This paper investigates the boundary regularity of asymptotically hyperbolic Einstein metrics with smooth Weyl curvature, establishing higher regularity of conformal compactifications under certain smoothness conditions.

Contribution

It extends boundary regularity results for asymptotically hyperbolic Einstein metrics by linking Weyl curvature smoothness to improved conformal compactification regularity.

Findings

Conformal compactifications are $C^{m+2,eta}$ up to the boundary under specified conditions.

Weyl curvature smoothness in $C^{m,eta}$ enhances boundary regularity of metrics.

The method follows Anderson’s approach to boundary regularity in Einstein metrics.

Abstract

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,\alpha}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order near boundary have conformal compactifications that are $C^{m+2,\alpha}$ up to the boundary when Weyl curvature is in $C^{m,\alpha}$ and the boundary metric is in $C^{m+2,\alpha}$ where $m\geq 3.$