Skip to main content
Log in

The Kähler Ricci flow on Fano surfaces (I)

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Suppose {(M, g(t)), 0 ≤ t < ∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricci flow on toric Fano surface, whose initial metric has toric symmetry. In particular, such a Kähler Ricci flow must converge to a Kähler Ricci soliton metric. Therefore we give a new Ricci flow proof of the existence of Kähler Ricci soliton metrics on toric Fano surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Anderson, M.T.: On the topology of complete manifolds of non-negative Ricci curvature. Topology 28 (1989)

  2. Anderson M.T.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc. 2(3), 455–490 (1989)

    Article  MATH  Google Scholar 

  3. Cao H.: Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bando S., Kasue A., Nakajima H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97(2), 313–349 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bando S.: Bubbling out of Einstein manifolds. Tohoku Math. J. 42(2), 205–216 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen X., LeBrun C., Weber B.: On conformally Kähler, Einstein manifolds. J. Am. Math. Soc. 21, 1137–1168 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen X., Wang B.: Remarks on Kähler Ricci flow. J. Geom. Anal. 20(2), 335–353 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, X., Tian, G.: Ricci flow on Kähler–Einstein surfaces, Invent Math. 147(3), 487–544

  9. Chen X., Tian G.: Ricci flow on Kähler–Einstein manifolds. Duke Math. J. 131(1), 17–73 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)

    MATH  MathSciNet  Google Scholar 

  11. Hamilton R.S.: A compactness property for solutions of the Ricci flow. Am. J. Math. 117, 545–572 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  12. Koiso, N.: On rotationally symmetric Hamilton’s equation for Kähler–Einstein metrics, recent topics in differential and analytic geometry, 327–337, Adv. Stud. Pure Math., 18-I, Academic Press, Boston (1990)

  13. Kronheimer P.B.: The construction of ALE spaces as Hyper-Kähler quotients. J. Differ. Geom. 29, 65–683 (1989)

    Google Scholar 

  14. Phong D.H., Song J., Sturm J., Weinkove B.: The Kähler–Ricci Flow with positive bisectional curvature. Invent. Math. 173(3), 651–665 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Phong D.H., Song J., Sturm J., Weinkove B.: The Kähler–Ricci Flow and the \({\bar{\partial}}\) operator on vector fields. J. Differ. Geom. 81(3), 631–647 (2009)

    MATH  MathSciNet  Google Scholar 

  16. Ruan W., Zhang Y., Zhang Z.: Bounding sectional curvature along a Kähler–Ricci flow. Commun. Contemp. Math. 11(6), 1067–1077 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sesum, N.: Compactness results for the Kähler–Ricci flow. arXiv: math.DG/0707.2974

  18. Shi W.: Deforming the metric on complete Riemannian manifolds. J. Differ. Geom. 30(1), 223–301 (1989)

    MATH  Google Scholar 

  19. Shi W.: Ricci deformation of the metric on complete noncompact Riemannian manifolds. J. Differ. Geom. 30(2), 303–394 (1989)

    MATH  Google Scholar 

  20. Sesum N., Tian G.: Bounding scalar curvature and diameter along the Kähler–Ricci Flow (after Perelman) and some applications. J. Inst. Math. Jussieu 7(3), 575–587 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tian G.: On Calabi’s conjecture for complex complex surfacws with c 1 > 0. Commun. Math. Phys. 112(1), 175–203 (1987)

    Article  MATH  Google Scholar 

  22. Yau S.T., Tian G.: Kähler–Einstein metrics on complex surfacws with c 1 > 0. Commun. Math. Phys. 112(1), 175–203 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, X., Zhu, X.: Kähler–Ricci solitons on toric manifolds with positive first Chern class, Adv. Math. 188(1), 87–103

  24. Yau S.: On the Ricci curvatre of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)

    Article  MATH  Google Scholar 

  25. Zhu, X.: Kähler–Ricci flow on a toric manifold with positive first Chern class. arXiv: math/0703486

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bing Wang.

Additional information

Partially supported by an NSF grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, X., Wang, B. The Kähler Ricci flow on Fano surfaces (I). Math. Z. 270, 577–587 (2012). https://doi.org/10.1007/s00209-010-0813-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-010-0813-3

Keywords

Navigation