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Perturbative solutions to the extended constant scalar curvature equations on asymptotically hyperbolic manifolds


Author: Erwann Delay
Journal: Proc. Amer. Math. Soc. 137 (2009), 2293-2298
MSC (2000): Primary 35J50, 58J05, 35J70, 35J60, 35Q75
DOI: https://doi.org/10.1090/S0002-9939-09-09703-2
Published electronically: February 20, 2009
MathSciNet review: 2495262
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Abstract: The extended constant scalar curvature equations is a particular case of the conformal contraint equations introduced by H. Friedrich. It was first studied by A. Butscher in an asymptotically flat setting. We prove the local existence of solutions to the extended constant scalar curvature equations near some asymptotically hyperbolic Einstein metrics. This gives a new local construction of asymptotically hyperbolic metrics with constant scalar curvature.


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  • 1. M.T. Anderson, P.T. Chruściel, and E. Delay, Non-trivial, static, geodesically complete vacuum space-times with a negative cosmological constant. II: $ n\ge 5$, Proceedings of the Strasbourg Meeting on AdS-CFT correspondence (Berlin, New York) (O. Biquard and V. Turaev, eds.), IRMA Lectures in Mathematics and Theoretical Physics, vol. 8, de Gruyter, 2005, pp. 165-204. MR 2160871 (2007b:53160)
  • 2. L. Andersson and P.T. Chruściel, Solutions of the constraint equations in general relativity satisfying ``hyperboloidal boundary conditions'', Dissert. Math. 355 (1996), 100 pp. MR 1405962 (97e:58217)
  • 3. L. Andersson and V. Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Annales Henri Poincaré 4 (2003), 1-34. MR 1967177 (2004c:58060)
  • 4. A. Butscher, Exploring the conformal constraint equations, The Conformal Structure of Space-Time, Lect. Notes in Phys., 604, Springer, Berlin, 2002, pp. 195-222. MR 2007930 (2004j:83009)
  • 5. -, Perturbative solutions of the extended constraint equations in general relativity, Commun. Math. Phys. 272 (2007), no. 1, 1-23. MR 2291799 (2008c:83006)
  • 6. A. Chruściel and H. Friedrich (eds.), The Einstein equations and the large scale behavior of gravitational fields (50 years of the Cauchy problem in general relativity), Birkhäuser-Verlag, Basel-Boston-Berlin, 2004. MR 2098911 (2005f:83001)
  • 7. R. Gicquaud, De l'équation de prescription de courbure scalaire aux équations de contrainte en relativité générale sur une variété asymptotiquement hyperbolique (2008), arXiv:0802.3279.
  • 8. C.R. Graham and J.M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), 186-225. MR 1112625 (92i:53041)
  • 9. J.M. Lee, Fredholm operators and Einstein metrics on conformally compact manifolds, Memoirs AMS 183 (2006), no. 864. MR 2252687 (2007m:53047)
  • 10. R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Diff. Geom. 28 (1988), 309-339. MR 961517 (89i:58005)

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Additional Information

Erwann Delay
Affiliation: Institut de Mathématiques et Modélisation de Montpellier, UMR 5149 CNRS, Université Montpellier II, Place Eugène Bataillon, 34095 Montpellier cedex 5, France
Address at time of publication: Laboratoire d’Analyse Non linéaire et Géométrie (EA2151), Faculté des Sciences, 33 rue Louis Pasteur, F-84000 Avignon, France
Email: Erwann.Delay@univ-avignon.fr

DOI: https://doi.org/10.1090/S0002-9939-09-09703-2
Keywords: Asymptotically hyperbolic manifolds, general relativity, constraint equations, symmetric 2-tensor, asymptotic behavior
Received by editor(s): March 17, 2008
Published electronically: February 20, 2009
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2009 American Mathematical Society

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