The scalar curvature equation on 2- and 3-spheres

  • Sun-Yung A. Chang
  • Matthew J. Gursky
  • Paul C. Yang


In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.

Mathematics subject classifications

35J60 58G03 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Aubin, T.: Meileures constantes dans le théorème d inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal.32, 149–179 (1979)Google Scholar
  2. [BC]
    Bahri, A., Coron, J.M.: The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal.95, 106–172 (1991)Google Scholar
  3. [Bo]
    Bourguignon, J.P.: Invariants integraux fonctionnels pour des equations aux derivees partielles d'origine geometrique. (Lect. Notes Math., vol. 1209, pp. 100–108) Berlin Heidelberg New York: Springer 1985Google Scholar
  4. [BE]
    Bianchi, G., Egnell, H.: An ODE approach to the equation Δu+Ku 205-01=0, in Rn (preprint)Google Scholar
  5. [CGS]
    Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math.42, 271–289 (1989)Google Scholar
  6. [CY1]
    Chang, S.Y.A., Yang, P.C.: Prescribing Gaussian curvature onS 2. Acta Math.159, 215–259 (1987)Google Scholar
  7. [CY2]
    Chang, S.Y.A., Yang, P.C.: Conformal deformation of metric onS 2. J. Differ. Geom.27, 259–296 (1988)Google Scholar
  8. [CY3]
    Chang, S.Y.A., Yang, P.C.: A perturbation result in prescribing scalar curvature onS n. Duke Math.64, 27–69 (1991)Google Scholar
  9. [ChD]
    Chen, W.-X., Ding, W.-Y.: Scalar curvature onS 2. Trans. AMS303, 369–382 (1987)Google Scholar
  10. [ChS]
    Cheng, K.S., Smoller, J.: Preprint.Google Scholar
  11. [ES]
    Escobar, J., Schoen, R.: Conformal metrics with prescribed scalar curvature. Invent. Math.86, 243–254 (1986)Google Scholar
  12. [G]
    Gursky, M.: Compactness of conformal metrics with integral bounds on curvature. Thesis, Cal Tech., 1991.Google Scholar
  13. [Ha]
    Han, Z-C.: Prescribing Gaussian curvature onS 2. Duke Math.61, 679–703 (1990)Google Scholar
  14. [Ho]
    Hong, C.: A best constant and the Gaussian curvature. Proc. AMS97, 737–747 (1986)Google Scholar
  15. [KW]
    Kazdan, J., Warner, F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math.101, 317–331 (1975)Google Scholar
  16. [L]
    Li, Y.Y.: On −Δu=K(x)u5 in R3. Commun. Pure Appl. Math.46, 303–340 (1993)Google Scholar
  17. [M]
    Moser, J.: On a non-linear problem. In: Peixoto, M. (ed.) Differential geometry dynamical Systems. New York: Academic Press 1973Google Scholar
  18. [N]
    Nirenberg, L.: Nonlinear functional analysis. NYU Lecture NotesGoogle Scholar
  19. [O]
    Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. Math. Soc. Japan14, 333–340 (1962)Google Scholar
  20. [On]
    Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys.86, 321–326 (1982)Google Scholar
  21. [SZ]
    Schoen, R., Zhang, D.: Private communicationGoogle Scholar
  22. [XY]
    Xu, X.W., Yang, P.: Remarks on prescribing Gaussian curvature on S2. Trans. AMS. (to appear)Google Scholar
  23. [Z]
    Zhang, D.: New result on geometric variational problems. Thesis, Stanford University (1990)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Sun-Yung A. Chang
    • 1
  • Matthew J. Gursky
    • 2
  • Paul C. Yang
    • 3
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations