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Low regularity Poincaré-Einstein metrics


Authors: Eric Bahuaud and John M. Lee
Journal: Proc. Amer. Math. Soc. 146 (2018), 2239-2252
MSC (2010): Primary 53C21; Secondary 35B65, 35J57, 35J70, 53C25
DOI: https://doi.org/10.1090/proc/13903
Published electronically: December 18, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of a $ C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $ -1$ plus terms of order $ e^{-2r}$ where $ r$ is the distance from any fixed compact set. This metric has no $ C^2$ conformal compactification.


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  • [1] Paul T. Allen, James Isenberg, John M. Lee, and Iva Stavrov Allen, Weakly asymptotically hyperbolic manifolds, Comm. Anal. Geom., to appear.
  • [2] Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429-461. MR 794369, https://doi.org/10.2307/1971181
  • [3] Eric Bahuaud, Intrinsic characterization for Lipschitz asymptotically hyperbolic metrics, Pacific J. Math. 239 (2009), no. 2, 231-249. MR 2457230, https://doi.org/10.2140/pjm.2009.239.231
  • [4] Eric Bahuaud and Romain Gicquaud, Conformal compactification of asymptotically locally hyperbolic metrics, J. Geom. Anal. 21 (2011), no. 4, 1085-1118. MR 2836592, https://doi.org/10.1007/s12220-010-9179-3
  • [5] Romain Gicquaud, Conformal compactification of asymptotically locally hyperbolic metrics II: weakly ALH metrics, Comm. Partial Differential Equations 38 (2013), no. 8, 1313-1367. MR 3169748, https://doi.org/10.1080/03605302.2013.795966
  • [6] C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186-225. MR 1112625, https://doi.org/10.1016/0001-8708(91)90071-E
  • [7] Xue Hu, Jie Qing, and Yuguang Shi, Regularity and rigidity of asymptotically hyperbolic manifolds, Adv. Math. 230 (2012), no. 4-6, 2332-2363. MR 2927372, https://doi.org/10.1016/j.aim.2012.04.013
  • [8] John M. Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183 (2006), no. 864, vi+83. MR 2252687, https://doi.org/10.1090/memo/0864
  • [9] Yuguang Shi and Gang Tian, Rigidity of asymptotically hyperbolic manifolds, Comm. Math. Phys. 259 (2005), no. 3, 545-559. MR 2174416, https://doi.org/10.1007/s00220-005-1370-1

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Additional Information

Eric Bahuaud
Affiliation: Department of Mathematics, Seattle University, 901 12th Avenue, Seattle, Washington 98122
Email: bahuaude@seattleu.edu

John M. Lee
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: johnmlee@uw.edu

DOI: https://doi.org/10.1090/proc/13903
Received by editor(s): February 19, 2017
Received by editor(s) in revised form: July 16, 2017
Published electronically: December 18, 2017
Additional Notes: This work was supported by a grant from the Simons Foundation (#426628, Eric Bahuaud)
Communicated by: Guofang Wei
Article copyright: © Copyright 2017 American Mathematical Society

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