Boundary regularity for asymptotically hyperbolic metrics with smooth Weyl curvature

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 $m \geq 3$ , a $C^{m, α}$ asymptotically hyperbolic metric that satisfies the asymptotic Einstein condition $∥ E (g_{+}) ∥_{g_{+}} = ∥ {Ric}_{g_{+}} + n g_{+} ∥_{g_{+}} = o (ρ^{2})$ is in fact $C^{m + 2, α}$ , provided that its Weyl curvature is $C^{m, α}$ and the metric on the boundary that represents its conformal infinity is $C^{m + 2, α}$ .

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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

Authors

Xiaoshang Jin
Beijing International Center for Mathematical Research Peking University Haidian District Beijing China

Vol. 301, No. 2, 2019

Xiaoshang Jin

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors