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Prescribed scalar curvature on then-sphere

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Abstract

This paper considers the prescribed scalar curvature problem onS n forn>-3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We then show that forn=3 this is the only blow up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed scalar curvature problem onS 3.

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References

  1. A. Bahri, J.M. Coron: The scalar curvature problem on the standard three-dimensional sphere. J. Funct. Anal.95 (1991) 106–172

    Google Scholar 

  2. A. Chang, M. Gursky, P. Yang: The scalar curvature equation on the 2- and 3-sphere. Calc. Var.1 (1993) 205–229

    Google Scholar 

  3. A. Chang, P. Yang: Conformal deformation of metrics onS 2. J. Differ. Geom.27 (1988) 256–296

    Google Scholar 

  4. Z.C. Han: Prescribing Gaussian curvature onS 2. Duke Math. J.61 (1990) 679–703

    Google Scholar 

  5. Z.C. Han: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. Henri Poincaré anal. non linéaire8 (1991) 159–175

    Google Scholar 

  6. J. Kazdan, F. Warner: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math.101 (1975) 317–331

    Google Scholar 

  7. Y. Li: Prescribing scalar curvature onS n and related problems, Part I (to appear in J. Differ. Eqns.)

  8. R. Schoen: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. Lecture Notes in Mathematics, No. 1365 (ed. M. Giaquinta) 1988, 120–154

  9. R. Schoen: A report on some recent progress on nonlinear problems in geometry. Surveys in Differ. Geom.1 (1991) 201–241

    Google Scholar 

  10. D. Zhang: New results on geometric variational problems. Stanford Dissertation, 1990

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Authors and Affiliations

  1. Mathematics Department, Stanford University, 94305, Stanford, CA, USA

    Richard Schoen

  2. Mathematics Department, Johns Hopkins University, 21218, Baltimore, MD, USA

    Dong Zhang

Authors
  1. Richard Schoen
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  2. Dong Zhang
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Schoen, R., Zhang, D. Prescribed scalar curvature on then-sphere. Calc. Var 4, 1–25 (1996). https://doi.org/10.1007/BF01322307

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  • DOI: https://doi.org/10.1007/BF01322307

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