Skip to main content
SpringerLink
Log in

On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein's field equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The regularity of the solutions to the Yamabe Problem is considered in the case of conformally compact manifolds and negative scalar curvature. The existence of smooth hyperboloidal initial data for Einstein's field equations is demonstrated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andersson, L.: Elliptic systems on manifolds with asymptotically negative curvature. Submitted at Indiana Univ. Math. J. 1990

  2. Andersson, L., Chruściel, P. T.: In preparation

  3. Aviles, P., McOwen, R. C.: Complete conformal metrics with negative scalar curvature in compact Riemannian manifolds. Duke Math. J.56, 395–398 (1988)

    Google Scholar 

  4. Aviles, P., McOwen, R. C.: Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds. J. Differ. Geom.27, 225–239 (1988)

    Google Scholar 

  5. Besse, A. L.: Einstein manifolds, Vol.10 (Ergebnisse der Math.3 Folge) Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  6. Friedrich, H.: On static and radiative spacetimes. Commun. Math. Phys.119, 51–73 (1988)

    Google Scholar 

  7. Friedrich, H.: On the global existence and the asymptotic behaviour of solutions to the Einstein-Maxwell-Yang-Mills equation. J. Differ. Geom.34, 275–345 (1991)

    Google Scholar 

  8. Graham, C. R., Lee, J. M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math.87, 186–225 (1991)

    Google Scholar 

  9. Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. In: Contributions to analysis. Ahlfors, L. V. et al. (eds.) New York: Academic Press 1974

    Google Scholar 

  10. Mazzeo, R.: Hodge cohomology of negatively curved manifolds. Technical report, 1986

  11. Mazzeo, R.: The Hodge cohomology of conformally compact metrics. J. Differ. Geom.28, 309–339 (1988)

    Google Scholar 

  12. Mazzeo, R.: Regularity for the singular Yamabe problem. Stanford University preprint, 1990

  13. Mazzeo, R., Melrose, R. B.: Meromorphic extension of the resolvent on complete spaces with asymptotically negative curvature. J. Funct. Anal.75, 260–310 (1987)

    Google Scholar 

  14. Melrose, R. B.: Transformation of boundary value problems. Acta Math.147, 149–236 (1981)

    Google Scholar 

  15. Penrose, R.: Zero rest-mass fields including gravitation: asymptotic behaviour. Proc. R. Soc. Lond.A284, 159–203 (1965)

    Google Scholar 

  16. Schoen, R. M.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom.20, 479–495 (1984)

    Google Scholar 

Download references

Author information

Author notes

  1. Piotr T. Chruściel

    Present address: Institute of Mathematics of the Polish Academy of Sciences, Warsaw

Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Lars Andersson

  2. Center for Mathematics and its Applications, Australian National University, T2601, Canberra, AC, Australia

    Piotr T. Chruściel

  3. Max-Planck Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, W-8046, Garching bei München, FRG

    Helmut Friedrich

Authors
  1. Lars Andersson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr T. Chruściel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Helmut Friedrich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Supported in part by NFR, the Swedish Academy of Sciences and the Gustavsson Foundation

Rights and permissions

Reprints and permissions

About this article

Cite this article

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. Commun.Math. Phys. 149, 587–612 (1992). https://doi.org/10.1007/BF02096944

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02096944

Keywords

Search

Navigation