Formal power series for asymptotically hyperbolic Bach-flat metrics
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Formal power series for asymptotically hyperbolic Bach-flat metrics Aghil Alaee1,2 · Eric Woolgar3 Received: 24 August 2019 / Revised: 21 September 2020 / Accepted: 3 October 2020 © Springer Nature B.V. 2020
Abstract It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bachflat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant Q-curvature. This allows us to determine the free data at conformal infinity and to select those choices that lead to Einstein metrics. The asymptotically hyperbolic mass is part of that free data, in contrast to the pure Einstein case. Higher-dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4dimensional case, but for a generalized Bach equation suited to the Fefferman–Graham technique, we are able to obtain a relatively complete result illustrating an interesting splitting of the free data into low-order “Dirichlet” and high-order “Neumann” pairs. Keywords Bach tensor · Poincaré-Einsten manifolds · Asymptotically hyperbolic manifolds · Conformal gravity Mathematics Subject Classification 53Z05 · 58Axx · 53Cxx
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Eric Woolgar [email protected] Aghil Alaee [email protected] ; [email protected]
1
Department of Mathematics and Computer Science, Clark University, Worcester, MA 01610, USA
2
Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
3
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
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A. Alaee, E. Woolgar
1 Introduction In seminal work, Fefferman and Graham [17,18] studied formal series solutions of the Einstein equation for asymptotically hyperbolic metrics expanded about conformal infinity. This led to the identification of data for the singular boundary value problem for these metrics, the discovery of obstructions to power series solutions, and ultimately the discovery of new conformal invariants for the conformal boundary. It also laid the groundwork for holography within the AdS/CFT correspondence. More recently, Gover and Waldron [19] and Graham [20] have performed similar analyses for a scalar geometric PDE problem, a singular boundary value problem for the Yamabe equation. In 3-dimensions, this problem was solved in [7] as part of the construction of hyperboloidal initial data for the Einstein equations on spacetime. Albin [1] has announced an analysis of asymptotically hyperbolic formal series solutions of the Euler–Lagrange equations of Lovelock actions in arbitrary dimensions. Here we study the question of formal series expansions for
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