Boundary Estimate of Asymptotically Hyperbolic Einstein Manifolds of Even Dimension

TL;DR

This paper establishes improved boundary regularity estimates for asymptotically hyperbolic Einstein manifolds in even dimensions, linking boundary smoothness to interior metric regularity.

Contribution

It extends previous results by showing that under weaker initial regularity assumptions, the AHE metric attains higher boundary regularity, and provides estimates for the Yamabe compactification.

Findings

AHE metric is $C^{m, ext{ extalpha}}$ conformally compact under certain regularity conditions.

Improved boundary regularity results compared to Helliwell's work.

Provides estimates for the Yamabe compactification metric.

Abstract

In this paper, we study the finite boundary regularity and estimates of an asymptotically hyperbolic Einstein manifold in even dimension $n+1.$ We show that if the initial compactification is $C^{n-1}$ and the $(n-3)$-th derivative of its scalar curvature is H\"older continuous, then the AHE metric is $C^{m,\alpha}$ conformally compact provided the boundary metric is $C^{m,\alpha}$. This is an improvement of Helliwell's result. We also provide an estimate of the Yamabe compactification metric in the new structure.