Annales Henri Poincaré

, Volume 18, Issue 8, pp 2789–2814 | Cite as

Smoothly Compactifiable Shear-Free Hyperboloidal Data is Dense in the Physical Topology

  • Paul T. AllenEmail author
  • Iva Stavrov Allen


We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Hölder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.


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We thank James Isenberg and John M. Lee for helpful conversations.


  1. 1.
    Allen, P.T., Isenberg, J., Lee, J.M., Allen, I.S.: The shear-free condition and constant-mean-curvature hyperboloidal initial data. Class. Quantum Gravity 33(11), 115015 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allen, P.T., Isenberg, J., Lee, J.M., Allen, I.S.: Weakly asymptotically hyperbolic manifolds. Comm. Anal. Geom. arXiv:1506.03399
  3. 3.
    Andersson, L., Chruściel, P.T.: On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of scri. Comm. Math. Phys. 161(3), 533–568 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andersson, L., Chruściel, P.T.: Solutions of the constraint equations in general relativity satisfying “hyperboloidal” boundary conditions. Diss. Math. 355, 100 (1996)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Andersson, L., Chruściel, P.T., Friedrich, H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Comm. Math. Phys. 149(3), 587–612 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Choquet-Bruhat, Y.: General Relativity and the Einstein Equations. Oxford Mathematical Monographs. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  7. 7.
    Chruściel, P.T., Delay, E., Lee, J.M., Skinner, D.N.: Boundary regularity of conformally compact Einstein metrics. J. Differ. Geom. 69(1), 111–136 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Friedrich, H.: Cauchy problems for the conformal vacuum field equations in general relativity. Comm. Math. Phys. 91(4), 445–472 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Friedrich, H.: On static and radiative space-times. Comm. Math. Phys. 119(1), 51–73 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Isenberg, J., Lee, J.M., Allen, I.S.: Asymptotic gluing of asymptotically hyperbolic solutions to the Einstein constraint equations. Ann. Henri Poincaré 11(5), 881–927 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lee, J.M.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Amer. Math. Soc. 183 (864), vi+83 (2006)Google Scholar
  12. 12.
    Mazzeo, R.: Elliptic theory of differential edge operators I. Comm. Partial Differ. Equ. 16(10), 1615–1664 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesMSC 110, Lewis and Clark CollegePortlandUSA

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