The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1080–1097 | Cite as

Convergence of the Calabi Flow on Toric Varieties and Related Kähler Manifolds

  • Hongnian HuangEmail author


Let X be a toric manifold with a Delzant polytope P. We show that if (X,P) is analytic uniform K-stable and the curvature is uniformly bounded along the Calabi flow, then the modified Calabi flow converges to an extremal metric exponentially fast. By assuming that the curvature is uniformly bounded along the Calabi flow, we prove a conjecture of Donaldson and a conjecture of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman.


Calabi flow Toric manifolds Extremal metrics K-stability 

Mathematics Subject Classification

53C44 53C55 53D20 



The author would like to thank Vestislav Apostolov and Gábor Székelyhidi for many stimulating discussions. He is also grateful to the consistent support of Professor Xiuxiong Chen, Pengfei Guan, and Paul Gauduchon. He would like to thank Si Li, Jeff Streets, and Valentino Tosatti for their interest in this work.


  1. 1.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry I: general theory. J. Differ. Geom. 73, 359–412 (2006) zbMATHGoogle Scholar
  2. 2.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry III: extremal metrics and stability. Invent. Math. 173, 547–601 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.W.: Extremal Kähler metrics on projective bundles over a curve. Adv. Math. 227, 2385–2424 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Berman, R.: A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics. arXiv:1011.3976
  5. 5.
    Calabi, E.: Extremal Kähler metric. In: Yau, S.T. (ed.) Seminar of Differential Geometry. Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton (1982) Google Scholar
  6. 6.
    Calabi, E.: Extremal Kähler metric, II. In: Chavel, I., Farkas, H.M. (eds.) Differential Geometry and Complex Analysis, pp. 95–114. Springer, Berlin (1985) CrossRefGoogle Scholar
  7. 7.
    Calabi, E., Chen, X.X.: Space of Kähler metrics and Calabi flow. J. Differ. Geom. 61(2), 173–193 (2002) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Chen, X.X.: Calabi flow in Riemann surfaces revisited: a new point of view. Int. Math. Res. Not. 2001(6), 275–297 (2001) CrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, X.X., He, W.Y.: On the Calabi flow. Am. J. Math. 130(2), 539–570 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chen, X.X., He, W.Y.: The Calabi flow on Kähler surface with bounded Sobolev constant–(I). Math. Ann. 354(1), 227–261 (2012). arXiv:0710.5159 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Chen, X.X., He, W.Y.: The Calabi flow on toric Fano surface. Math. Res. Lett. 17(2), 231–241 (2010). arXiv:0807.3984 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, B.H., Li, A.M., Sheng, L.: Uniform K-stability for extremal metrics on toric varieties. arXiv:1109.5228
  13. 13.
    Chrusciél, P.T.: Semi-global existence and convergence of solutions of the Robison-Trautman (2-dimensional Calabi) equation. Commun. Math. Phys. 137, 289–313 (1991) CrossRefzbMATHGoogle Scholar
  14. 14.
    Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Donaldson, S.K.: Conjectures in Kähler geometry. In: Strings and Geometry. Clay Math. Proc., vol. 3, pp. 71–78. Am. Math. Soc., Providence (2004) Google Scholar
  16. 16.
    Donaldson, S.K.: Interior estimates for solutions of Abreu’s equation. Collect. Math. 56, 103–142 (2005) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Donaldson, S.K.: Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Donaldson, S.K.: Kähler geometry on toric manifolds, and some other manifolds with large symmetry. In: Handbook of Geometric Analysis. No. 1. Adv. Lect. Math. (ALM), vol. 7, pp. 29–75. International Press, Somerville (2008) Google Scholar
  19. 19.
    Donaldson, S.K.: Extremal metrics on toric surfaces: a continuity method. J. Differ. Geom. 79(3), 389–432 (2008) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Donaldson, S.K.: Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal. 19(1), 83–136 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Donaldson, S.K.: b-Stability and blow-ups. arXiv:1107.1699
  22. 22.
    Feng, R.J., Huang, H.N.: The global existence and convergence of the Calabi flow on \(\mathbb{C}^{n} = \mathbb{Z}^{n} + i \mathbb{Z}^{n}\). J. Funct. Anal. 263(4), 1129–1146 (2012). doi: 10.1016/j.jfa.2012.05.017 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Huang, H.N.: On the extension of the Calabi flow on toric varieties. Ann. Glob. Anal. Geom. 40(1), 1–19 (2011). arXiv:1101.0638 CrossRefzbMATHGoogle Scholar
  24. 24.
    Huang, H.N.: Toric surface, K-stability and Calabi flow. arXiv:1207.5964
  25. 25.
    Huang, H.N., Zheng, K.: Stability of Calabi flow near an extremal metric. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 11(1), 167–175 (2012). arXiv:1007.4571 zbMATHMathSciNetGoogle Scholar
  26. 26.
    Raza, A.A.: Scalar curvature and multiplicity-free actions. Ph.D. thesis, Imperial College London (2005) Google Scholar
  27. 27.
    Streets, J.: The long time behavior of fourth-order curvature flows. Calc. Var. Partial. Differ. Equ. 46(1–2), 39–54 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Struwe, M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1(2), 247–274 (2002) zbMATHMathSciNetGoogle Scholar
  29. 29.
    Székelyhidi, G.: Optimal test-configurations for toric varieties. J. Differ. Geom. 80, 501–523 (2008) zbMATHGoogle Scholar
  30. 30.
    Székelyhidi, G.: Filtrations and test-configurations. arXiv:1111.4986
  31. 31.
    Székelyhidi, G.: Extremal metrics and K-stability. Ph.D. thesis, Imperial college Google Scholar
  32. 32.
    Tian, G.: Kähler-Einstein metrics of positive scalar curvature. Invent. Math. 130, 1–57 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Tosatti, V.: Kähler-Ricci flow on stable Fano manifolds. J. Reine Angew. Math. 640, 67–84 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Yau, S.T.: Review of Kähler-Einstein metrics in algebraic geometry. Isr.l Math. Conf. Proc., Bar-Ilan Univ. 9, 433–443 (1996) Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.CMLSÉcole PolytechniquePalaiseauFrance

Personalised recommendations