The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 902–916 | Cite as

Dynamic Instability of \(\mathbb {CP}^N\) Under Ricci Flow

  • Dan Knopf
  • Nataša Šešum


The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kröncke [18] that complex projective space with its canonical Fubini–Study metric is dynamically unstable under Ricci flow in all complex dimensions \(N\ge 2\). The unstable perturbation is not Kähler. This provides a counterexample to a well-known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kähler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may have expected.


Ricci flow Dynamic instability Complex projective space 

Mathematics Subject Classification




Funding was provided by Division of Mathematical Sciences (Grant No. DMS-1056387).


  1. 1.
    Angenent, S., Knopf, D.: Precise asymptotics of the Ricci flow neckpinch. Comm. Anal. Geom. 15(4), 773–844 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bamler, R.H.: Stability of hyperbolic manifolds with cusps under Ricci flow. Adv. Math. 263, 412–467 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété Riemannienne. Lecture Notes in Mathematics, vol. 194. Springer, New York (1971)CrossRefzbMATHGoogle Scholar
  4. 4.
    Besse, A.L.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cao, H.D., Hamilton, R.S., Ilmanen, T.: Gaussian densities and stability for some Ricci solitons. arXiv:math/0404165v1 (2004)
  6. 6.
    Cao, H.D., He, C.: Linear stability of Perelman’s \(\nu \)-entropy on symmetric spaces of compact type. J. Reine Angew. Math. 709, 229–246 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cao, H.D., Zhu, M.: On second variation of Perelman’s Ricci shrinker entropy. Math. Ann. 353(3), 747–763 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dai, X., Wang, X., Wei, G.: On the stability of Riemannian manifold with parallel spinors. Invent. Math. 161(1), 151–176 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dai, X., Wang, X., Wei, G.: On the variational stability of Kähler-Einstein metrics. Comm. Anal. Geom. 15(4), 669–693 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Feldman, M., Ilmanen, T., Ni, L.: Entropy and reduced distance for Ricci expanders. J. Geom. Anal. 15(1), 49–62 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guenther, C., Isenberg, J., Knopf, D.: Stability of the Ricci flow at Ricci-flat metrics. Comm. Anal. Geom. 10(4), 741–777 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hamilton, R.S.: The formation of singularities in the Ricci flow. In: Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7—136, International Press, Cambridge, MA (1995)Google Scholar
  14. 14.
    Haslhofer, R.: Perelman’s \(\lambda \)-functional and the stability of Ricci-flat metrics. Calc. Var. Partial Differ. Equ. 45(3–4), 481–504 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Haslhofer, R., Müller, R.: Dynamical stability and instability of Ricci-flat metrics. Math. Ann. 360(1–2), 547–553 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Isenberg, J., Knopf, D., Šešum, N.: Non-Kähler Ricci flow singularities that converge to Kähler–Ricci solitons. arXiv:1703.02918
  17. 17.
    Knopf, D.: Convergence and stability of locally \(\mathbb{R}^N\)-invariant solutions of Ricci flow. J. Geom. Anal. 19(4), 817–846 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kröncke, K.: Stability of Einstein metrics under Ricci flow. Comm. Anal. Geom. arXiv:1312.2224v2 (2013)
  19. 19.
    Li, H., Yin, H.: On stability of the hyperbolic space form under the normalized Ricci flow. Int. Math. Res. Not. 15, 2903–2924 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
  21. 21.
    Šešum, N.: Linear and dynamical stability of Ricci-flat metrics. Duke Math. J. 133(1), 1–26 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schnürer, O.C., Schulze, F., Simon, M.: Stability of Euclidean space under Ricci flow. Comm. Anal. Geom. 16(1), 127–158 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Schnürer, O.C., Schulze, F., Simon, M.: Stability of hyperbolic space under Ricci flow. Comm. Anal. Geom. 19(5), 1023–1047 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tian, G., Zhu, X.: Convergence of the Kähler-Ricci flow on Fano manifolds. J. Reine Angew. Math. 678, 223–245 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Williams, M.B., Haotian, W.: Dynamical stability of algebraic Ricci solitons. J. Reine Angew. Math. 713, 225–243 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wu, H.: Stability of complex hyperbolic space under curvature-normalized Ricci flow. Geom. Dedicata 164, 231–258 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ye, R.: Ricci flow, Einstein metrics and space forms. Trans. Am. Math. Soc. 338(2), 871–896 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.University of Texas at AustinAustinUSA
  2. 2.Rutgers UniversityNew BrunswickUSA

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