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A sphere theorem on locally conformally flat even-dimensional manifolds

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Abstract

In this paper, we prove that a closed even-dimensional manifold which is locally conformally flat with positive scalar curvature, positive Euler characteristic and which satisfies some additional condition on its curvature is diffeomorphic to the sphere or projective space.

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References

  1. Besse A.L.: Einstein Manifolds. Springer, Berlin (1987)

    MATH  Google Scholar 

  2. Carron G., Herzlich M.: Conformally flat manifolds with non-negative Ricci curvature. Compos. Math. 142, 798–810 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang S.-Y.A., Han Z.-C., Yang P.C.: Classification of singular radial solutions to the k-Yamabe problem on annular domains. J. Differ. Equ. 216(2), 482–501 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen S.: Local estimates for some fully nonlinear elliptic equations. Int. Math. Res. Notices 55, 3403–3425 (2005)

    Article  Google Scholar 

  5. Evans L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)

    Article  MATH  Google Scholar 

  6. Gonzáles M.: Singular sets of a class of locally conformally flat manifolds. Duke Math. J. 129(3), 551–572 (2005)

    Article  MathSciNet  Google Scholar 

  7. Guan P., Viaclovsky J., Wang G.: Some properties of the Schouten tensor and applications to conformal geometry. Trans. Am. Math. Soc. 355, 925–933 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guan P., Lin C.S., Wang G.: Application of the method of moving planes to conformally invariant equations. Math. Z. 247(1), 1–19 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guan P., Lin C.S., Wang G.: Schouten tensor and some topological properties. Commun. Anal. Geom. 13(5), 887–902 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Gursky M.J.: Locally conformally flat four- and six-manifolds of positive scalar curvature and positive Euler characteristic. Indiana Univ. Math. J. 43(3), 747–774 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gursky M.J., Viaclovsky J.: A fully nonlinear equation on four-manifolds with positive scalar curvature. J. Differ. Geom. 63(1), 131–154 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Hebey E., et Vaugon M.: Un theoreme de pincement integral sur la courbure concirculaire en geometrie conforme. C. R. Acad. Sci. Paris 316, 483–488 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Hebey E., et Vaugon M.: Effective L p pinching for the concircular curvature. J. Geom. Anal. 6, 531–553 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Krylov N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat. 47(1), 75–108 (1983)

    MathSciNet  Google Scholar 

  15. Viaclovsky J.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101(2), 283–316 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Giovanni Catino

  2. Institut Fourier, UMR 5582, 100 rue des Maths, 38402, Saint Martin d’Hères Cedex, France

    Zindine Djadli

  3. SISSA, Via Beirut, 2-4, 34014, Trieste, Italy

    Cheikh Birahim Ndiaye

Authors
  1. Giovanni Catino
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  2. Zindine Djadli
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  3. Cheikh Birahim Ndiaye
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Correspondence to Giovanni Catino.

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Catino, G., Djadli, Z. & Ndiaye, C.B. A sphere theorem on locally conformally flat even-dimensional manifolds. manuscripta math. 136, 237–247 (2011). https://doi.org/10.1007/s00229-011-0443-z

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  • DOI: https://doi.org/10.1007/s00229-011-0443-z

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