Skip to main content
SpringerLink
Log in

Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space

  • Original Article
  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bartnik, R.: Existence of maximal surfaces in asymptotically flat spacetimes. Commun. Math. Phys. 94, 155–175 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bartnik, R.: Maximal surfaces and general relativity, Geometry and partial differential equations, 2nd Miniconf., Canberra/Aust. 1986, Proc. Cent. Math. Anal. Aust. Natl. Univ. 12, 24–49 (1987)

    MathSciNet  Google Scholar 

  3. Bayard, P.: Dirichlet problem for space-like hypersurfaces with prescribed scalar curvature in \({\mathcal{R}}^{n,1}\). Calc. Var. 18, 1–30 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. 2. Wiley Interscience Publishers (1962)

  5. Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math. 104(2), 407–419 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  6. Delanoë, Ph.: Dirichlet problem for the equation of prescribed Lorentz-Gauss curvature. Ukr. Math. J. 42(12), 1538–1545 (1990)

    Article  MATH  Google Scholar 

  7. Delanoë, Ph.: Personal communication. Arch (2005)

  8. Guan, B.: The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature. Trans. Amer. Math. Soc. 350, 4955–4971 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, second edition, Revised Third Printing. Springer-Verlag (1998)

  10. Li, A.M.: Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space. Arch. Math. 64(6), 534–551 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lin, M., Trudinger, N.S.: On some inequalities for elementary symmetric functions. Bull. Austral. Math. Soc. 50, 317–326 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  12. Potter, A.J.B.: An elementary version of the Leray-Schauder theorem. J. London Math. Soc. 5(2), 414–416 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Schnürer, O.: The Dirichlet problem for Weingarten hypersurfaces in Lorentzian manifolds. Math. Z. 242, 159–181 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Treibergs, A.: Entire Spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66, 39–56 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  15. Urbas, J.: The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space. Calc. Var. 18, 307–316 (2003)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Física y Matemáticas, U.M.S.N.H. Ciudad Universitaria, CP. 58040 Morelia, Michoacan, Mexico

    Pierre Bayard

Authors
  1. Pierre Bayard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Bayard.

Additional information

Mathematics Subject Classification (2000) 35J60, 53C50

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bayard, P. Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space. Calc. Var. 26, 245–264 (2006). https://doi.org/10.1007/s00526-005-0367-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-005-0367-0

Keywords

Search

Navigation