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The prescribed scalar curvature problem for metrics with unit total volume

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Abstract

We solve the modified Kazdan–Warner problem of finding metrics with prescribed scalar curvature and unit total volume.

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References

  1. Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55(3), 269–296 (1976)

    MathSciNet  MATH  Google Scholar 

  2. Bourguignon, J.P.: Une stratification de l’espace des structures riemanniennes. Compos. Math. 30, 1–41 (1975)

    MathSciNet  MATH  Google Scholar 

  3. Corvino, J., Eichmair, M., Miao, P.: Deformation of scalar curvature and volume. Math. Ann. 357(2), 551–584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kazdan, J.L., Warner, F.A.: A direct approach to the determination of Gaussian and scalar curvature functions. Invent. Math. 28, 227–230 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kazdan, J.L., Warner, F.W.: Scalar curvature and conformal deformation of Riemannian structure. J. Differ. Geom. 10, 113–134 (1975)

    MathSciNet  MATH  Google Scholar 

  6. Kobayashi, O.: Scalar curvature of a metric with unit volume. Math. Ann. 279(2), 253–265 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kobayashi, O.: Scalar curvature of spheres. Math. Z. 200(2), 273–277 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Koiso, N.: A decomposition of the space \({\cal M}\) of Riemannian metrics on a manifold. Osaka J. Math. 16(2), 423–429 (1979)

    MathSciNet  MATH  Google Scholar 

  9. Koiso, N.: Einstein metrics and complex structures. Invent. Math. 73(1), 71–106 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schoen, R.M.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math., vol. 1365, pp. 120–154. Springer, Berlin (1989)

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Acknowledgments

The author wishes to express his gratitude to O. Kobayashi for suggesting the problem and for many stimulating conversations, and to K. Akutagawa, N. Koiso, and N. Otoba for useful discussions on various aspects of this work. He also gratefully acknowledges the many helpful suggestions of the anonymous referee.

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Authors and Affiliations

  1. Department of Mathematics, Osaka University, Toyonaka, Osaka , 560-0043, Japan

    Shinichiroh Matsuo

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  1. Shinichiroh Matsuo
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Correspondence to Shinichiroh Matsuo.

Additional information

This work was supported by Grant-in-Aid for Young Scientists (B) 25800045.

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Matsuo, S. The prescribed scalar curvature problem for metrics with unit total volume. Math. Ann. 360, 675–680 (2014). https://doi.org/10.1007/s00208-014-1052-4

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  • DOI: https://doi.org/10.1007/s00208-014-1052-4

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