Multiple Point Blowup Phenomenon in Scalar Curvature Equations on Spheres of Dimension Greater Than Three

  • Yanyan Li
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 20)


Let (S n,g o) be the standard n—sphere. The following question was raised by L. Nirenberg. Which function K on S2 is the Gauss curvature of a metric g on S 2 conformally equivalent to g o? Naturally one may ask a similar question in higher dimensional case, namely, which function K on S n is the scalar curvature of a metric g on Sn conformally equivalent to g o?


Scalar Curvature Morse Index Morse Function Constant Scalar Curvature High Dimensional Case 
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  1. [A]
    T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations. Springer-Verlag, New York, 1982.MATHGoogle Scholar
  2. [BC]
    A. Bahri, J.M. Coron, The scalar-curvature problem on standard three-dimensional sphere, J. of Func. Anal., 95 (1991), 106 – 172.MathSciNetMATHCrossRefGoogle Scholar
  3. [BLR]
    A. Bahri, Y.Y. Li, O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calculus of Variations and PDE’s 3 (1995), 67 – 93.MathSciNetMATHCrossRefGoogle Scholar
  4. [VGS]
    L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271 – 297.MathSciNetMATHCrossRefGoogle Scholar
  5. [CGY]
    S.Y. Chang, M.J. Gursky, P. Yang, The scalar curvature equation on 2- and 3-sphere, Calculus of Variations and PDE’s 1 (1993), 205 – 229.MathSciNetMATHCrossRefGoogle Scholar
  6. [CY]
    S.Y. Chang, P. Yang, Conformal deformations of metrics on S2, J. Diff. Geom. 27 (1988), 256 – 296.MathSciNetGoogle Scholar
  7. [CD]
    W.X. Chen, W. Ding, Scalar curvature on S2, Trans. Amer. Math. Soc. 303 (1987), 365 – 382.MathSciNetMATHGoogle Scholar
  8. [ES]
    J. Escobar, R. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 (1986), 243 – 254.MathSciNetMATHCrossRefGoogle Scholar
  9. [H]
    Z.C. Han, Prescribing Gaussian curvature on S2, Duke Math. J. 61 (1990), 679 – 703.MathSciNetMATHCrossRefGoogle Scholar
  10. [KW]
    J. Kazdan, F. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. of Math. 101 (1975), 317 – 331.MathSciNetMATHCrossRefGoogle Scholar
  11. [LI]
    Y.Y. Li, Prescribing scalar curvature on Snand related problems, Part I, J. Differential Equations, 120 (1995), 319–410.MathSciNetMATHCrossRefGoogle Scholar
  12. [L2]
    Y.Y. Li, Prescribing scalar curvature on Snand related problems, Part II: existence and compactness, Comm. Pure Appl. Math., 49 (1996).Google Scholar
  13. [M]
    J. Moser, On a nonlinear problem in differential geometry, Dynam-ical systems (M. Peixoto, ed.) Academic Press, New York, 1973.Google Scholar
  14. [S]
    R. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A symposium in honor of Manfredo Do Carmo (H.B. Lawson and K. Tenenblat, eds ), Wiley, 1991, 311–320.Google Scholar
  15. [Z]
    D. Zhang, New results on geometric variational problems, thesis, Stanford University, 1990.Google Scholar

Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Yanyan Li
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

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