Annales Henri Poincaré

, Volume 14, Issue 5, pp 1109–1134 | Cite as

Gluing Construction of Initial Data with Kerr–de Sitter Ends

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Abstract

We construct initial data sets which satisfy the vacuum constraint equations of General Relativity with positive cosmological constant. More precisely, we deform initial data with ends asymptotic to Schwarzschild–de Sitter to obtain non-trivial initial data with exactly Kerr–de Sitter ends. The method is inspired from Corvino’s gluing method. We obtain here a extension of a previous result for the time-symmetric case by Chruściel and Pollack (Ann H Poincaré 9(4):639–654, 2008). We also deal with the case of asymptotically Kerr–de Sitter initial data.

Keywords

Initial Data Constraint Equation Constraint Operator Spacelike Hypersurface Positive Cosmological Constant 
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References

  1. 1.
    Andersson L., Chruściel P.T.: On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of Scri. Commun. Math. Phys. 161, 533–568 (1994)ADSMATHCrossRefGoogle Scholar
  2. 2.
    Bartnik R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bartnik, R., Isenberg, J.: The constraint equations. In: Chruściel, P.T., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields: 50 Years of the Cauchy Problem in General Relativity, pp. 1–34. Birkhauser, Basel (2004)Google Scholar
  4. 4.
    Beig R., Chruściel P.T.: Killing initial data. Class. Quantum Grav. 14, A83–A92 (1997)MATHCrossRefGoogle Scholar
  5. 5.
    Carter B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968)ADSMATHCrossRefGoogle Scholar
  6. 6.
    Choquet-Bruhat Y.: Théorèmes d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Choquet-Bruhat Y., Geroch R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys. 14, 329–335 (1969)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Chruściel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr., no. 94 (2003)Google Scholar
  9. 9.
    Chruściel P.T., Delay E.: Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature. Commun. Anal. Geom. 17(2), 343–381 (2009)MATHGoogle Scholar
  10. 10.
    Chruściel P.T., Pacard F., Pollack D.: Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends II. Math. Res. Lett. 16(1), 157–164 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    Chruściel P.T., Pollack D.: Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends. Ann. H. Poincaré 9(4), 639–654 (2008)MATHCrossRefGoogle Scholar
  12. 12.
    Corvino J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214(1), 137–189 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Corvino J., Schoen R.M.: On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73(2), 185–217 (2006)MathSciNetMATHGoogle Scholar
  14. 14.
    Delay E.: Localized gluing of Riemannian metrics in interpolating their scalar curvature. Differ. Geom. Appl. 29(3), 433–439 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Delay, E.: Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications. Commun. PDE (2011)Google Scholar
  16. 16.
    Gibbons G.W., Hawking S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15(10), 2738–2751 (1977)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Gicquaud, R.: Étude de quelques problèmes d’analyse et de géométrie sur les variétés asymptotiquement hyperboliques. Ph.D.thesis, Université Montpellier 2 (2009)Google Scholar
  18. 18.
    Hawking, S.W., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, no. 1, Cambridge University Press, Cambridge (1973)Google Scholar
  19. 19.
    Humbert, E.: Relativité générale (d’après M. Vaugon) et quelques problèmes mathématiques qui en sont issus. arXiv: 1004.2402v4 [math.DG] (2010)Google Scholar
  20. 20.
    Kruskal M.D.: Maximal extension of Schwarzschild metric. Phys. Rev. 119, 1743–1745 (1960)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Maerten D.: Killing initial data revisited. J. Math. Phys. 45(7), 2594–2599 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Mazzeo R., Pollack D., Uhlenbeck K.: Moduli spaces of singular Yamabe metrics. J. Am. Math. Soc. 9(2), 303–344 (1996)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Michel B.: Geometric invariance of mass-like asymptotic invariants. J. Math. Phys. 52(5), 052504 (2011)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    O’Neill, B.: The Geometry of Kerr Black Holes. A.K. Peters, Wellesley (1995)Google Scholar
  25. 25.
    Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R.A., Nugent, P., Castro, P.G., Deustua, S., Fabbro, S., Goobar, A., Groom, D.E., Hook, I.M., Kim, A.G., Kim, M.Y., Lee, J.C., Nunes, N.J., Pain, R., Pennypacker, C.R., Quimby, R., Lidman, C., Ellis, R.S., Irwin, M., McMahon, R.G., Ruiz-Lapuente, P., Walton, N., Schaefer, B., Boyle, B.J., Filippenko, A.V., Matheson, T., Fruchter, A.S., Panagia, N., Newberg, H.J.M., Couch, W.J., The Supernova Cosmology Project: Measurements of Omega and Lambda from 42 High-Redshift Supernovae. ApJ 517, 565–586 (1999). arXiv: astro-ph/9812133Google Scholar
  26. 26.
    Schoen, R.M.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of Variations (Montecatini Terme, 1987). Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin (1989)Google Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik, (Albert-Einstein-Institut)GolmGermany

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