Skip to main content
SpringerLink
Log in

On the convergence of a modified Kähler–Ricci flow

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the convergence of a modified Kähler–Ricci flow defined by Zhou Zhang. We show that the modified Kähler–Ricci flow converges to a singular metric when the limit class is degenerate. This proves a conjecture of Zhang.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen X.X., Tian G.: Ricci flow on Kähler-Einstein surfaces. Invent. Math. 147(3), 487–544 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Demailly, J.-P.: Complex analytic and algebraic geometry, online book available on the webpage of J.-P. Demailly

  4. Dinew, S., Zhang, Z.: Stability of bounded solutions for degenerate complex Monge–Ampère equations, ArXiv:0711.3643

  5. Donaldson S.: Scalar curvature and projective embeddings I. J. Differential Geom 59(3), 479–522 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Eyssidieux P., Guedj V., Guedj V.: A Singular Kähler-Einstein metrics. J. Amer. Math. Soc 22(3), 607–639 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Kołodziej S.L: The complex Monge–Ampre equation. Acta Math. 180(1), 69–117 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kołodziej S.: The complex Monge–Ampère equation and pluripotential theory. Mem. Amer. Math. Soc. 178(840), pp. x+64 (2005)

    Google Scholar 

  10. Perelman, G.: Unpublished notes on the Kähler–Ricci flow

  11. Phong D.H., Sturm J.: On stability and the convergence of the Kähler–Ricci flow. J. Differential Geom. 72(1), 149–168 (2006)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Siu, Y.T.: Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, DMV Seminar, 8. Birkhäuser Verlag, Basel, 171 p. (1987)

  14. Song J., Tian G.: The Kähler–Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Song, J., Tian, G.: Canonical measures and Kähler–Ricci flow, ArXiv:0802.2570

  16. Song, J., Tian, G.: The Kähler–Ricci flow through singularities, ArXiv:0909.4898

  17. Song, J., Yuan, Y.: The Kähler–Ricci flow on Calabi–Yau varieties, In Preparation

  18. Tian G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tian, G.: Canonical metrics in Kähler geometry, Lectures in Mathematics. ETH Zürich, Birkhauser Verlag, Basel (2000)

  20. Tian G., Zhang Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chinese Ann. Math. Ser. B 27(2), 179–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tian G., Zhu X.: Convergence of Kähler–Ricci flow. J. Am. Math. Soc. 20(3), 675–699 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tosatti V.: Limits of Calabi–Yau metrics when the Kahler class degenerates. J. Eur. Math. Soc. (JEMS) 11(4), 755–776 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tsuji H.: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281, 123–133 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Weinkove B.: The Calabi–Yau equation on almost-Kähler four-manifolds. J. Differential Geom. 76(2), 317–349 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yau, S.-T.: Problem section Seminar on differential geometry, pp. 669–706, Ann. of Math. Stud., 102, Princeton Univ. Press, Princeton (1982)

  27. Zhang, Z.: Degenerate Monge–Ampère equations over projective manifolds, PhD thesis in MIT

  28. Zhang, Z.: On degenerate Monge–Ampère equations over closed Kähler manifolds, Int. Math. Res. Not. 2006, Art. ID 63640, 18 p.

  29. Zhang Z.: A Modified Kähler–Ricci Flow. Math. Ann. 345(3), 559–579 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Yuan Yuan

Authors
  1. Yuan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Yuan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yuan, Y. On the convergence of a modified Kähler–Ricci flow. Math. Z. 268, 281–289 (2011). https://doi.org/10.1007/s00209-010-0670-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-010-0670-0

Keywords

Search

Navigation