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Courbures scalaires des variétés d'invariant conforme négatif


Author: Antoine Rauzy
Journal: Trans. Amer. Math. Soc. 347 (1995), 4729-4745
MSC: Primary 53C21; Secondary 35J60
DOI: https://doi.org/10.1090/S0002-9947-1995-1321588-5
MathSciNet review: 1321588
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Abstract: In this paper, we are interested in the problem of prescribing the scalar curvature on a compact riemannian manifold of negative conformal invariant. We give a necessary and sufficient condition when the prescribed function $ f$ is nonpositive. When $ \sup(f) > 0$, we merely find a sufficient condition. This is the subject of the first theorem. In the second one, we prove the multiplicity of the solutions of subcritical (for the Sobolev imbeddings) elliptic equations. In another article [8], we will prove the multiplicity of the solutions of the prescribing curvature problem, i.e. for a critical elliptic equation.


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  • [1] A. Ambrosetti et P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. MR 0370183 (51:6412)
  • [2] Th. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl 55 (1976), 269-296. MR 0431287 (55:4288)
  • [3] -, Nonlinear analysis on manifolds--Monge-Ampère equations, Grundlehren der Math. Wissenschaften, vol. 252, Springer-Verlag, Berlin, 1982.
  • [4] J. L. Kazdan et F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom. 10 (1975), 113-134. MR 0365409 (51:1661)
  • [5] J. L. Kazdan, Prescribing the curvature of a Riemannian manifold, CBMS Regional Conf. Ser. in Math., no. 57, Amer. Math. Soc., Providence, RI, 1985; reprinted with corrections 1987. MR 787227 (86h:53001)
  • [6] T. Ouyang, On the positive solutions of semilinear equations $ \Delta u + \lambda u + h{u^p} = 0$ on compact manifolds. II, Indiana Univ. Math. J. 40 (1991), 1083-1141. MR 1129343 (92m:35017)
  • [7] A. Rauzy, Courbure scalaire prescrite, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 273-276. MR 1205197 (94f:53078)
  • [8] -, Multiplicité pour un problème de courbure scalaire prescrite (à paraître).
  • [9] J. L. Vazquez et L. Véron, Solutions positives d'équations elliptiques semi-linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris. Sér. I Math. 312 (1991), 811-815. MR 1108497 (92g:35066)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1321588-5
Article copyright: © Copyright 1995 American Mathematical Society

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