Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor

Abstract

In this paper, we classify three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function.

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  • 07 July 2020

    In the original article the formula chapter 4, just above Remark 4.2 in the paragraph starting with Case (2) the formula should read.

References

  1. 1.

    Bach, R.: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs. Math. Z. 9, 110–135 (1921)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194, 731–764 (2013)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci soliton. Trans. A.M.S. 364, 2377–2391 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cao, H.-D., Chen, Q.: On Bach-flat gradient shrinking Ricci solitons. Duke Math. J. 162, 1149–1169 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Cao, H.-D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach-flat gradient steady Ricci solitons. Calc. Var. 49, 125–138 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cao, H.-D., Sun, X., Zhang, Y.: On the structure of gradient Yamabe solitons. Math. Res. Lett. 19, 767–774 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Catino, G., Mantegazza, C., Mazzieri, L.: On the global structure of conformal gradient solitons with nonnegative Ricci tensor. Commun. Contemp. Math. 14, 12 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow, Graduate Studies in Mathematics, vol. 77. American Mathematical Society, Providence (2006)

    Google Scholar 

  9. 9.

    Daskalopoulos, P., Sesum, N.: The classification of locally conformally flat Yamabe solitons. Adv. Math. 240, 346–369 (2013)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Hamilton, R.: Lectures on geometric flows, unpublished (1989)

  11. 11.

    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hsu, S.Y.: A note on compact gradient Yamabe solitons. J. Math. Anal. Appl. 388(2), 725–726 (2012)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Huang, G., Li, H.: On a classification of the quasi Yamabe gradient solitons. Meth. and Appl. of Anal. 21, 379–390 (2014)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Japan. 19, 205–214 (1967)

    MathSciNet  Article  Google Scholar 

  15. 15.

    O’Neill, B.: Semi-Riemannian Geometry With Applications to Relativity. Academic Press, Cambridge (1983)

    Google Scholar 

  16. 16.

    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002). arXiv:math.DG/0211159

  17. 17.

    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). arXiv:math.DG/0307245

  18. 18.

    Perelman, G.: Ricci flow with surgery on three-manifolds (2003). arXiv:math.DG/0303109

  19. 19.

    Thurston, W.P.: Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. 124, 203–246 (1986)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The work was done while the author was visiting the Department of Mathematics of Texas A & M University-Commerce as a Visiting Scholar, and he is grateful to the department and the university for the hospitality he had received during the visit. He also would like to thank the referee for his/her valuable comments and suggestions.

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Correspondence to Shun Maeta.

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The author is partially supported by the Grant-in-Aid for Young Scientists, Nos. 15K17542 and 19K14534 Japan Society for the Promotion of Science, and JSPS Overseas Research Fellowships 2017-2019 No. 70.

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Maeta, S. Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. Ann Glob Anal Geom (2020). https://doi.org/10.1007/s10455-020-09722-9

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Keywords

  • Yamabe solitons
  • Cotton tensor
  • Bach tensor
  • Scalar curvature

Mathematics Subject Classification

  • 53C21
  • 53C25
  • 53C20