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On Compact Riemannian Manifolds with Harmonic Weyl Curvature

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Abstract

We give some rigidity theorems for an n-dimensional (\(n\ge 4\)) compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant \(\sigma _2\) curvature. Moreover, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant \(\sigma _2\) curvature is isometric to a quotient of the round \(\mathbb {S}^4\).

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References

  1. Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998)

    Book  Google Scholar 

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

    Book  Google Scholar 

  3. Bour, V.: Fourth order curvature flows and geometric applications. arXiv:1012.0342

  4. Branson, T.: Kato constants in Riemannian geometry. Math. Res. Lett. 7, 245–261 (2000)

    Article  MathSciNet  Google Scholar 

  5. Carron, G.: Some old and new results about rigidity of critical metric. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13, 1091–1113 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Catino, G.: On conformally flat manifolds with constant positive scalar curvature. Proc. Am. Math. Soc. 144, 2627–2634 (2016)

    Article  MathSciNet  Google Scholar 

  7. Catino, G.: Integral pinched shrinking Ricci solitons. Adv. Math. 303, 279–294 (2016)

    Article  MathSciNet  Google Scholar 

  8. Chang, S.A., Gursky, M.J., Yang, P.C.: A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes Études Sci. 98, 105–143 (2003)

    Article  MathSciNet  Google Scholar 

  9. Derdziński, A.: On compact Riemannian manifolds with harmonic curvature. Math. Ann. 259, 144–152 (1982)

    Article  MathSciNet  Google Scholar 

  10. Fu, H.P.: Four manifolds with postive Yamabe constant. Pac. J. Math. 296, 79–104 (2018)

    Article  Google Scholar 

  11. Fu, H.P.: On compact manifolds with harmonic curvature and positive scalar curvature. J. Geom. Anal. 27, 3120–3139 (2017)

    Article  MathSciNet  Google Scholar 

  12. Fu, H.P., Peng, J.K.: Conformally flat Riemannian manifolds with finite \(L^p\)-norm curvature. Annali di Matematica 196, 1903–1912 (2017)

    Article  MathSciNet  Google Scholar 

  13. Fu, H.P., Xiao, L.Q.: Some \(L^p\) rigidity results for complete manifolds with harmonic curvature. Potential Anal. 48(2018), 239–255 (2017)

    Google Scholar 

  14. Fu, H.P., Xiao, L.Q.: Einstein manifolds with finite \(L^p\)-norm of the Weyl curvature. Differ. Geom. Appl. 53, 293–305 (2017)

    Article  Google Scholar 

  15. Fu, H.P., Xu, G.B., Tao, Y.Q.: Some remarks on Bach-flat manifolds with positive constant scalar curvature. Colloq. Math. 155, 187–196 (2019). https://doi.org/10.4064/cm7358-2-2018

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Hu, Z.J., Li, H., Simon, U.: Schouten curvature functions on locally conformally flat Riemannian manifolds. J. Geom. 88, 75–100 (2008)

    Article  MathSciNet  Google Scholar 

  19. Huisken, G.: Ricci deformation of the metric on a Riemannian manifold. J. Differ. Geom. 21, 47–62 (1985)

    Article  MathSciNet  Google Scholar 

  20. Itoh, M., Satoh, H.: Isolation of the Weyl conformal tensor for Einstein manifolds. Proc. Jpn. Acad. A Math. Sci. 78, 140–142 (2002)

    Article  MathSciNet  Google Scholar 

  21. Jack, I., Parker, L.: Linear independence of renormalisation counterterms in curved space–times of arbitrary dimensionality. J. Math. Phys. 28, 1137–1139 (1987)

    Article  MathSciNet  Google Scholar 

  22. Kim, S.: Rigidity of noncompact complete manifolds with harmonic curvature. Manuscr. Math. 135, 107–116 (2011)

    Article  MathSciNet  Google Scholar 

  23. Lee, M.J., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc. 17, 37–91 (1987)

    Article  MathSciNet  Google Scholar 

  24. Li, A.M., Zhao, G.S.: Isolation phenomena for Riemannian manifolds whose Ricci curvature tensor are parallel. Acta Math. Sci. Ser. A Chin. Ed. 37, 19–24 (1994)

    MATH  Google Scholar 

  25. Li, H.: Global rigidity theorems of hypersurfaces. Ark. Mat. 35, 327–351 (1997)

    Article  MathSciNet  Google Scholar 

  26. Pigola, S., Rigoli, M., Setti, A.G.: Some characterizations of space-forms. Trans. Am. Math. Soc. 359, 1817–1828 (2007)

    Article  MathSciNet  Google Scholar 

  27. Simon, U.: A further method in global differential geometry. Abh. Math. Semin. Univ. Hambg. 44, 52–69 (1976)

    Article  MathSciNet  Google Scholar 

  28. Singer, M.: Positive Einstein metrics with small \(L^{n/2}\)-norm of the Weyl tensor. Differ. Geom. Appl. 2, 269–274 (1992)

    Article  Google Scholar 

  29. Tachibana, S.: A theorem on Riemannian manifolds of positive curvature operator. Proc. Jpn. Acad. 50, 301–302 (1974)

    Article  MathSciNet  Google Scholar 

  30. Tran, H.: On closed manifolds with harmonic Weyl curvature. Adv. Math. 322, 861–891 (2017)

    Article  MathSciNet  Google Scholar 

  31. Xu, H.W., Zhao, E.T.: \(L^p\) Ricci curvature pinching theorems for conformally flat Riemannian manifolds. Pac. J. Math. 245, 381–396 (2010)

    Article  Google Scholar 

  32. Zhou, Z.R.: Inequalities of Simons type and gaps for Yang–Mills fields. Ann. Glob. Anal. Geom. 48, 223–232 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to Professor Haizhong Li for his guidance and constant support.

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Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, 330031, People’s Republic of China

    Hai-Ping Fu

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Hui-Ya He

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  1. Hai-Ping Fu
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  2. Hui-Ya He
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Correspondence to Hui-Ya He.

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Supported by National Natural Science Foundations of China #11761049, Jiangxi Province Natural Science Foundation of China #20171BAB201001.

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Fu, HP., He, HY. On Compact Riemannian Manifolds with Harmonic Weyl Curvature. Results Math 74, 77 (2019). https://doi.org/10.1007/s00025-019-0994-y

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