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 3, a Cm,α asymptotically hyperbolic metric that satisfies the asymptotic Einstein condition E(g+)g+ = Ricg+ + ng+g+ = o(ρ2) is in fact Cm+2,α, provided that its Weyl curvature is Cm,α and the metric on the boundary that represents its conformal infinity is Cm+2,α.

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