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On locally conformally flat manifolds with finite total Q-curvature

Abstract

In this paper, we study the ends of a locally conformally flat complete manifold with finite total Q-curvature. We prove that for such a manifold, the integral of the Q-curvature equals an integral multiple of a dimensional constant \(c_n\), where \(c_n\) is the integral of the Q-curvature on the unit n-sphere. It provides further evidence that the Q-curvature on a locally conformally flat manifold controls geometry as the Gaussian curvature does in two dimension.

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Notes

  1. 1.

    By the Sard Theorem, for almost all \(r_i^j\) this property is true.

  2. 2.

    See also [16, Page 245].

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Acknowledgements

The second author is grateful to Alice Chang and Paul Yang for discussions and interest in the work. She would also like to thank Matt Gursky for interest to the work and suggestions. Both authors are very grateful to the referees for his suggestions to improve the presentation of the paper.

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Correspondence to Yi Wang.

Additional information

Zhiqin Lu is partially supported by NSF grant DMS-1510232, and Yi Wang is partially supported by NSF grants DMS-1547878 and DMS-1612015.

Communicated by A. Malchiodi.

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Lu, Z., Wang, Y. On locally conformally flat manifolds with finite total Q-curvature. Calc. Var. 56, 98 (2017). https://doi.org/10.1007/s00526-017-1189-6

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Mathematics Subject Classification

  • Primary 53A30
  • Secondary 53C21