A priori estimates for Donaldson’s equation over compact Hermitian manifolds



In this paper we prove a priori estimates for Donaldson equation’s
$$\begin{aligned} \omega \wedge (\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n-1} =e^{F}(\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n}, \end{aligned}$$
over a compact complex manifold \(X\) of complex dimension \(n\), where \(\omega \) and \(\chi \) are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove (Asian J. Math. 14:19–40, 2010).

Mathematics Subject Classification (2000)

Primary 53C55 Secondary 32W20 35J60 58J05 



The author would like to thank Kefeng Liu, Valentino Tosatti, Xiaokui Yang for useful discussions on Donaldson’s equation, the complex Monge-Ampère equation and geometric flows. The author thanks referees’s helpful suggestions.


  1. 1.
    Chen, X.: On the lower bound of the Mabuchi energy and its application. Internat. Math. Res. Notices 12. 607–623 (2000)Google Scholar
  2. 2.
    Chen, X.: A new parabolic flow in Kähler manifolds. Commun. Anal. Geom. 12(4), 837–852 (2004)CrossRefzbMATHGoogle Scholar
  3. 3.
    Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3(1), 1–15 (1999)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Fang, H.: Lai, Mijia: On the geometric flows solving Kählerian inverse \(\sigma _{k}\) equations. Pacific J. Math. 258(2), 291–304 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Fang, H., Lai, M.: Convergence of general inverse \(\sigma _{k}\)-flow on Kähler manifolds with Calabi Ansatz. arXiv: 1203.5253. In: Transactions of the American Mathematical Society (2013, to appear)Google Scholar
  6. 6.
    Fang, H., Lai, M., Ma, X.: On a class of fully nonlinear flow in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fang, H., Lai, M., Song, J., Weinkove, B.: The \(J\)-flow on Kähler surfaces: a boundary case. arXiv: 1204.4068 (2012)Google Scholar
  8. 8.
    Gill, M.: Convergence of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. Commun. Anal. Geom. 19(2), 277–303 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Guan, B.: Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. arxiv: 1211.0181 (2012)Google Scholar
  10. 10.
    Guan B., Li. Q.: Complex Monge-Ampère equations and totally real submanifolds. Adv. Math. 225(3), 1185–1223 (2010)Google Scholar
  11. 11.
    Guan, B., Li, Q.: A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete Contin. Dyn. Syst. Ser. B 17(6), 1991–1999 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Guan, B., Li, Q.: The Dirichlet problem for a complex Monge-Ampère type equation on Hermitian manifolds. arXiv: 1210.5526 (2012)Google Scholar
  13. 13.
    Guan, B., Sun, W.: On a class of fully nonlinear elliptic equations on Hermitian manifolds. arXiv: 1301.5863 (2013)Google Scholar
  14. 14.
    Liu, K.-F., Yang, X.-K.: Geometry of Hermitian manifolds. Internat. J. Math. 23(6), 1250055 (2012)Google Scholar
  15. 15.
    Song, J., Weinkove, B.: On Donaldson’s flow of surfaces in a hyperkähler four-manifold. Math. Z. 256(4), 769–787 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Song, J., Weinkove, B.: On the convergence and singularities of the \(J\)-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61(2), 210–229 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Tosatti, V., Weinkove, B.: Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14(1), 19–40 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Tosatti, V., Weinkove, B.: The complex Monge-Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23(4), 1187–1195 (2010)Google Scholar
  19. 19.
    Tosatti, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern-Ricci form. arxiv: 1201.0312v2 (2012)Google Scholar
  20. 20.
    Tosatti, V., Weinkove, B.: The chern-Ricci flow on complex surfaces. arXiv:1209.2663 (2012)Google Scholar
  21. 21.
    Tosatti, V., Weinkove, B., Yang, X.: Collapsing of the Chern-Ricci flow on elliptic surfaces. arXiv: 1302.6545 (2013)Google Scholar
  22. 22.
    Weinkove, B.: Convergence of the \(J\)-flow on Kähler surfaces. Commun. Anal. Geom. 12(4), 949–965 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Weinkove, B.: On the \(J\)-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73(2), 351–358 (2006)Google Scholar
  24. 24.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Commun Pure. Appl. Math. 31(3) 339–411 (1978)Google Scholar
  25. 25.
    Zhang, X.: A priori estimates for complex Monge-Ampère equation on Hermitian manifolds. Int. Math. Res. Not. IMRN 19. 3814–3836 (2010)Google Scholar
  26. 26.
    Zhang, X., Zhang, X.: Regularity estimates of solutions of complex Monge-Ampère equations on Hermitian manifolds. J. Funct. Anal. 260(7), 2004–2026 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina

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