#### Vol. 301, No. 2, 2019

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Boundary regularity for asymptotically hyperbolic metrics with smooth Weyl curvature

### Xiaoshang Jin

Vol. 301 (2019), No. 2, 467–487
DOI: 10.2140/pjm.2019.301.467
##### Abstract

We study the regularity of asymptotically hyperbolic metrics in general dimensions. By carefully constructing harmonic coordinates near the boundary at infinity, a method pioneered by Anderson, we show that, for $m\ge 3$, a ${C}^{m,\alpha }$ asymptotically hyperbolic metric that satisfies the asymptotic Einstein condition $\parallel E\left({g}_{+}\right){\parallel }_{{g}_{+}}=\parallel {Ric}_{{g}_{+}}\phantom{\rule{0.3em}{0ex}}+n{g}_{+}{\parallel }_{{g}_{+}}=o\left({\rho }^{2}\right)$ is in fact ${C}^{m+2,\alpha }\phantom{\rule{0.3em}{0ex}}$, provided that its Weyl curvature is ${C}^{m,\alpha }$ and the metric on the boundary that represents its conformal infinity is ${C}^{m+2,\alpha }\phantom{\rule{0.3em}{0ex}}$.

##### Keywords
regularity, asymptotically hyperbolic, harmonic coordinates, Einstein, Weyl curvature
##### Mathematical Subject Classification 2010
Primary: 53A30, 53C21, 53C25, 58J05