Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons

  • Huai-Dong CaoEmail author
  • Xin Cui


In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay \(\lim _{x\rightarrow \infty } R(x)=0\).


Curvature estimate Steady Ricci solitons 

Mathematics Subject Classification

53C21 53C25 53C44 



We are grateful to Ovidiu Munteanu and Jiaping Wang for sending us their paper [25], and its early version in July 2014, which motivated us to consider curvature estimates for 4D steady solitons. The first author also would like to thank Ovidiu Munteanu for very helpful discussions; part of the work was carried out when the first author was visiting University of Macau, where he was partially supported by Science and Technology Development Fund (Macao S.A.R.) Grant FDCT/016/2013/A1 and the RDG010 Project of the University of Macau.


  1. 1.
    Brendle, S.: Rotational symmetry of self-similar solutions to the Ricci flow. Invent. Math. 194(3), 731–764 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brendle, S.: Rotational symmetry of Ricci solitons in higher dimensions. J. Differ. Geom. 97, 191–214 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
  4. 4.
    Buzano, M., Dancer, A.S., Gallaugher, M., Wang, M.: A family of steady Ricci solitons and Ricci-flat metrics, preprint (2014), arXiv:1309.6140v2 [math.DG]
  5. 5.
    Cao, H.-D.: Existence of gradient Kähler-Ricci solitons. In: Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), Peters, AK., Wellesley, MA., pp. 1–16, (1996)Google Scholar
  6. 6.
    Cao, H.D.: Recent Progress on Ricci Solitons, Recent Advances in Geometric Analysis, 1–38, Adv. Lect. Math. (ALM). International Press, Somerville, MA (2010)Google Scholar
  7. 7.
    Cao, H.-D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach-flat gradient steady Ricci solitons. Calc. Var. Partial Diff. Equ. 49(1–2), 125–138 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Amer. Math. Soc. 364, 2377–2391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cao, H.-D., Zhou, D.: On complete gradient shrinking Ricci solitons. J. Diff. Geom. 85(2), 175–186 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cao, H.-D., Zhu, X.-P.: A complete proof of the Poincaré and geometrization conjectures–application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10(2), 165–492 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Catino, G., Mantegazza, C.: Evolution of the Weyl tensor under the Ricci flow. Ann. Inst. Fourier. 61(4), 1407–1435 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Diff. Geom. 82, 363–382 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, X.X., Wang, Y.: On four-dimensional anti-self-dual gradient Ricci solitons, to appear in J. Geom. Anal. (arXiv: 1102.0358v2)
  14. 14.
    Chow, B., et al.: The Ricci flow: Techniques and Applications. Part I Geometric Aspects. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2007)Google Scholar
  15. 15.
    Chow, B., Lu, P., Yang, B.: Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons. C. R. Math. Acad. Sci. Paris 349(23–24), 1265–1267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dancer, A., Wang, M.: Some new examples of non-Kähler Ricci solitons. Math. Res. Lett. 16, 349–363 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dancer, A., Wang, M.: On Ricci solitons of cohomogeneity one. Ann. Glob. Anal. Geom. 39, 259–292 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fang, F., Li, X.-D., Zhang, Z.: Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature. Ann. Inst. Fourier (Grenoble) 59(2), 563–573 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Diff. Geom. 65, 169–209 (2003)CrossRefzbMATHGoogle Scholar
  20. 20.
    Fernndez-López, M., Garca-Río, E.: A sharp lower bound for the scalar curvature of certain steady gradient Ricci solitons. Proc. Amer. Math. Soc. 141(6), 2145–2148 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hamilton, R.S.: The Ricci flow on surfaces. Contemp. Math. 71, 237–261 (1988)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hamilton, R.S.: The formation of singularities in the Ricci flow. Surv. Diff. Geom. 2, 7–136 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ivey, T.: New examples of complete Ricci solitons. Proc. AMS 122, 241–245 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Munteanu, O., Sesum, N.: On gradient Ricci solitons. J. Geom. Anal. 23(2), 539–561 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Munteanu, O., Wang, J.: Geometry of shrinking Ricci solitons, preprint (2014), arXiv:1410.3813v1
  26. 26.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159
  27. 27.
    Perelman, G.: Ricci flow with surgery on three manifolds, arXiv:math/0303109
  28. 28.
    Petersen, P., Wylie, W.: Rigidity of gradient Ricci solitons. Pacific J. Math. 241, 329–345 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wei, G., Wylie, W.: Comparison geometry for the Bakry–Emery Ricci tensor. J. Diff. Geom. 83, 377–405 (2009)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wei, G., Wu, P.: On volume growth of gradient steady Ricci solitons. Pacific J. Math. 265(1), 233–241 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wu, P.: On the potential function of gradient steady Ricci solitons. J. Geom. Anal. 23, 221–228 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA

Personalised recommendations