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 $\parallel E\left(g_{+}\right){\parallel }_{g_{+}}=\parallel {Ric}_{g_{+}}+n{g}_{+}{\parallel }_{g_{+}}=o\left({\rho }^2\right)$ is in fact $C^{m+2,\alpha }$, provided that its Weyl curvature is $C^{m,\alpha }$ and the metric on the boundary that represents its conformal infinity is $C^{m+2,\alpha }$.

Keywords

regularity, asymptotically hyperbolic, harmonic coordinates, Einstein, Weyl curvature

Mathematical Subject Classification 2010

Primary: 53A30, 53C21, 53C25, 58J05

Milestones

Received: 16 January 2018

Revised: 8 September 2018

Accepted: 8 January 2019

Published: 24 October 2019