Boundary Regularity for Asymptotically Hyperbolic Metrics with Smooth Weyl Curvature

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 $m\ge 3.$