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
By the Sard Theorem, for almost all \(r_i^j\) this property is true.
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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Communicated by A. Malchiodi.
Zhiqin Lu is partially supported by NSF grant DMS-1510232, and Yi Wang is partially supported by NSF grants DMS-1547878 and DMS-1612015.
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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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DOI: https://doi.org/10.1007/s00526-017-1189-6