The Parabolic Flows for Complex Quotient Equations



We apply the parabolic flow method to solving complex quotient equations on closed Kähler manifolds. We study the parabolic equation and prove the convergence. As a result, we solve the complex quotient equations.


Parabolic flow Complex quotient equation \(\mathcal {C}\)-subsolution 

Mathematics Subject Classification

58J05 58J35 53C55 



The author is very grateful to Bo Guan for his encouragement and helpful conversations.


  1. 1.
    Aubin, T.: Équations du type Monge-Ampère sur les variétés kählériennes compactes (French). Bull. Sci. Math. 2(102), 63–95 (1978)MATHGoogle Scholar
  2. 2.
    Blocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier (Grenoble) 55(5), 1735–1756 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blocki, Z.: On uniform estimate on Calabi–Yau theorem. Sci. China Ser. A 48, 244–247 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blocki, Z.: On geodesics in the space of Kähler metrics. Advanced Lectures in Mathematics, vol. 21, pp. 3–20. International Press (2012)Google Scholar
  5. 5.
    Calabi, E.: The space of Kähler metrics. In: Proceedings of the ICM, Amsterdam 1954, vol. 2, pp. 206–207. North-Holland, Amsterdam (1956)Google Scholar
  6. 6.
    Cao, H.-D.: Deformation of Kähler matrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math. 81, 359–372 (1985)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, X.-X.: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Not. 12, 607–623 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cherrier, P.: Equations de Monge–Ampère sur les variétés hermitiennes compactes. Bull. Sci. Math. 111, 343–385 (1987)MathSciNetMATHGoogle Scholar
  9. 9.
    Collins, T.C., Székelyhidi, G.: Convergence of the \(J\)-flow on toric manifolds. J. Differ. Geom. 107(1), 47–81 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dinew, S., Kolodziej, S.: Liouville and Calabi-Yau type theorems for complex Hessian equations. Am. J. Math. 139(2), 403–415 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3, 1–16 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Evans, L.C.: Classical solutions of fully nonlinear, convex, secondorder elliptic equations. Commun. Pure Appl. Math. 35, 333–363 (1982)CrossRefMATHGoogle Scholar
  13. 13.
    Fang, H., Lai, M.-J., Ma, X.-N.: On a class of fully nonlinear flows in Kähler geometry. J. Reine Angew. Math. 653, 189–220 (2011)MathSciNetMATHGoogle Scholar
  14. 14.
    Gill, M.: Convergence of the parabolic complex Monge–Ampère equation on compact Hermitian manifolds. Commun. Anal. Geom. 19(2), 277–304 (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Gill, M.: Long time existence of the (n-1)-plurisubharmonic flow. Preprint. arXiv:1410.6958
  16. 16.
    Guan, B., Li, Q.: Complex Monge–Ampeère equations and totally real submanifolds. Adv. Math. 225, 1185–1223 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Guan, B., Sun, W.: On a class of fully nonlinear elliptic equations on Hermitian manifolds. Calc. Var. PDE 54(1), 901–916 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hou, Z.-L.: Complex Hessian equation on Kähler manifolds. Int. Math. Res. Not. 2009(16), 3098–3111 (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Hou, Z.-L., Ma, X.-N., Wu, D.-M.: A second order estimate for complex Hessian equations on a compact Kähler manifold. Math. Res. Lett. 17(3), 547–561 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izvestiya Ross. Akad. Nauk. SSSR 46, 487–523 (1982)MATHGoogle Scholar
  21. 21.
    Lejmi, M., Székelyhidi, G.: The J-flow and stability. Adv. Math. 274, 404–431 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, H.-Z., Shi, Y.-L., Yao, Y.: A criterion for the properness of the K-energy in a general Kähler class. Math. Ann. 361(1), 135–156 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Phong, D.H., Tô, D.T.: Fully non-linear parabolic equations on compact Hermitian manifolds. Preprint. arXiv:1711.10697
  24. 24.
    Song, J., Weinkove, B.: On the convergence and singularities of the J-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61, 210–229 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds. J. Geom. Anal. 26(3), 2459–2473 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Sun, W.: Parabolic complex Monge–Ampère type equations on closed Hermtian manifolds. Calc. Var. PDE 54(4), 3715–3733 (2015)CrossRefMATHGoogle Scholar
  27. 27.
    Sun, W.: On a class of fully nonlinear elliptic equations on closed Hermitian manifolds II: \(L^\infty \) estimate. Commun. Pure Appl. Math. 70(1), 172–199 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Székelyhidi, G.: Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differ. Geom. 109(2), 337–378 (2018)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tosatti, V., Wang, Y., Weinkove, B., Yang, X.-K.: \(C^{2,\alpha }\) estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc. Var. PDE 54(1), 431–453 (2015)CrossRefMATHGoogle Scholar
  30. 30.
    Tosatti, V., Weinkove, B.: Estimates for the complex Monge–Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14, 19–40 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Tosatti, V., Weinkove, B.: The complex Monge–Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23, 1187–1195 (2010)CrossRefMATHGoogle Scholar
  32. 32.
    Tosatti, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern-Ricci form. J. Differ. Geom. 99, 125–163 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Trudinger, N.S.: Fully nonlinear, uniformly elliptic equations under natural structure conditions. Trans. Am. Math. Soc. 278(2), 751–769 (1983)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tso, K.: On Aleksandrov–Bakel’man type maximum principle for second order parabolic equations. Commun. PDE’s 10(5), 543–553 (1985)CrossRefMATHGoogle Scholar
  35. 35.
    Wang, L.-H.: On the regularity theory of fully nonlinear parabolic equations. I. Commun. Pure Appl. Math. 45(1), 27–76 (1992)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang, L.-H.: On the regularity theory of fully nonlinear parabolic equations. II. Commun. Pure Appl. Math. 45(2), 141–178 (1992)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Weinkove, B.: Convergence of the J-flow on Kähler surfaces. Commun. Anal. Geom. 12, 949–965 (2004)CrossRefMATHGoogle Scholar
  38. 38.
    Weinkove, B.: On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differ. Geom. 73, 351–358 (2006)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. Commun. Pure Appl. Math. 31, 339–411 (1978)CrossRefMATHGoogle Scholar
  40. 40.
    Zhang, D.-K.: Hessian equations on closed Hermitian manifolds. Pac. J. Math. 291(2), 485–510 (2017)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhang, X.-W.: A priori estimate for complex Monge–Ampère equation on Hermitian manifolds. Int. Math. Res. Not. 2010, 3814–3836 (2010)MATHGoogle Scholar

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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

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