Rigidity of \(\kappa \)-noncollapsed steady Kähler–Ricci solitons

  • Yuxing Deng
  • Xiaohua ZhuEmail author


In this paper, we show that any n-dimensional \(\kappa \)-noncolla-psed steady (gradient) Kähler–Ricci soliton with nonnegative bisectional curvature must be flat. The result is an improvement to our former work in Deng and Zhu (Trans Am Math Soc 370(4):2855–2877, 2018).

Mathematics Subject Classification

Primary 53C25 Secondary 53C55 58J05 



The work was done partially when the second named author was visiting at the Mathematical Sciences Research Institute at Berkeley during the spring 2016 semester. He would like to thank her hospitality and the financial supports, NSF Grants DMS-1440140, and Simons Foundation. The authors also thank referees for valuable suggestions to the paper.


  1. 1.
    Bishop, R.L., Goldberg, S.I.: On the second cohomology group of a Kähler manifold of positive curvature. Proc. Am. Math. Soc. 16, 119–122 (1965)zbMATHGoogle Scholar
  2. 2.
    Bryant, R.L.: Gradient Kähler Ricci solitons. Géomtrié différentielle, physi-que mathématique, mathématiques et société. I. Astérisque (321), 51–97 (2008)Google Scholar
  3. 3.
    Cao, H.D.: Limits of solutions to the Kähler–Ricci flow. J. Diff. Geom. 45, 257–272 (1997)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, H.D.: On dimension reduction in the Kähler–Ricci flow. Comm. Anal. Geom. 12(1), 305–320 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, H.D., Chen, B.L., Zhu, X.P.: Recent developments on Hamilton’s Ricci flow. Surveys J. Diff. Geom. 12, 47–112 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cao, H.D., Chen, B.L., Zhu, X.P.: Ricci flow on compact Kähler manifolds with positive bisectional curvature. C.R. Math. Acad. Sci. Paris 337, 781–784 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chau, A., Tam, L.F.: On the complex structure of Kähler manifolds with nonnegative curvature. J. Diff. Geom. 73(3), 491–530 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chau, A., Tam, L.F.: Non-negatively curved Kähler manifolds with average quadratic curvature decay. Comm. Anal. Geom. 15(1), 121–146 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geom. 6, 119–128 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Diff. Geom. 17, 15–53 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, B.L.: Strong uniqueness of the Ricci flow. J. Diff. Geom. 82, 363–382 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chen, B.L., Zhu, X.P.: Volume growth and curvature decay of positively curved Kähler manifolds. Q. J. Pure Appl. Math. 1(1), 68–108 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, X.X., Sun, S., Tian, G.: A note on Kähler–Ricci soliton. Int. Math. Res. Not. 17, 3328–3336 (2009)zbMATHGoogle Scholar
  14. 14.
    Chow, B., etc.: The Ricci flow: Techniques and Applications, Part1: Geometric Aspects, Math. Surveys and Monographs, vol. 135, Amer. Math. Society., Providence, RI., (2007)Google Scholar
  15. 15.
    Deng, Y.X., Zhu, X.H.: Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature. Math. Z. 279(1–2), 211–226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Deng, Y.X., Zhu, X.H.: Asymptotic behavior of positively curved steady Ricci solitons. Trans. Am. Math. Soc. 370(4), 2855–2877 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hamilton, R.S.: Three manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hamilton, R.S.: Formation of singularities in the Ricci flow. Surveys Diff. Geom. 2, 7–136 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, G.: On the volume growth of Kähler manifolds with nonnegative bisectional curvature. J. Diff. Geom. 102(3), 485–500 (2016)CrossRefzbMATHGoogle Scholar
  20. 20.
    Morgan, J., Tian, G.: Ricci flow and the Poincaré conjecture, Clay Math. Mono., 3. Amer. Math. Soc., Providence, RI; Clay Mathematics Institute, Cambridge, MA (2007)Google Scholar
  21. 21.
    Ni, L.: A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature. J. Am. Math. Soc. 17, 909–946 (2004)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ni, L.: Ancient solutions to Kähler–Ricci flow. Math. Res. Lett. 12, 633–654 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ni, L., Tam, L.F.: Kähler–Ricci flow and the Poincaré–Lelong equation. Comm. Anal. Geom. 12(1–2), 111–141 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159
  25. 25.
    Rothaus, O.: Logarithmic Sobolev inequality and the spectrum of Schrödinger operators. J. Funct. Anal. 42, 110–120 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schoen, R., Yau, S.T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology, Vol. 1, International Press Publications, (1994)Google Scholar
  27. 27.
    Sesum, N.: Convergence of a Kähler–Ricci flow. Math. Res. Lett. 12, 623–632 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sesum, N., Tian, G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Juss. 7(3), 575–587 (2008)zbMATHGoogle Scholar
  29. 29.
    Shi, W.X.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Diff. Geom. 30, 223–301 (1989)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shi, W.X.: Ricci flow and the uniformization on complete noncompact Kähler manifolds. J. Diff. Geom. 45(1), 94–220 (1997)CrossRefzbMATHGoogle Scholar
  31. 31.
    Tam, L.F.: Uniformization of open nonnegatively curved Kähler manifolds in higher dimensions. In: Ji, L., Li, P., Schoen, R., Simon, L. (eds.), Handbook of Geometric analysis (Vol. I), Higher Education Press (2008)Google Scholar
  32. 32.
    Tian, G., Zhu, X.: Convergence of the Kähler–Ricci flow on Fano manifolds. J. Reine Angew. Math. 678, 223–245 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Wu, H.: On the de Rham decomposition Theorem. Illinois J. Math. 8, 291–311 (1964)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  2. 2.School of Mathematical Sciences and BICMRPeking UniversityBeijingChina

Personalised recommendations