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The Parabolic Flows for Complex Quotient Equations

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Abstract

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.

Keywords

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

Mathematics Subject Classification

58J05 58J35 53C55 

Notes

Acknowledgements

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

References

  1. 1.
    Aubin, T.: Équations du type Monge-Ampère sur les variétés kählériennes compactes (French). Bull. Sci. Math. 2(102), 63–95 (1978)zbMATHGoogle Scholar
  2. 2.
    Blocki, Z.: Weak solutions to the complex Hessian equation. Ann. Inst. Fourier (Grenoble) 55(5), 1735–1756 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blocki, Z.: On uniform estimate on Calabi–Yau theorem. Sci. China Ser. A 48, 244–247 (2005)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cherrier, P.: Equations de Monge–Ampère sur les variétés hermitiennes compactes. Bull. Sci. Math. 111, 343–385 (1987)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Collins, T.C., Székelyhidi, G.: Convergence of the \(J\)-flow on toric manifolds. J. Differ. Geom. 107(1), 47–81 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Donaldson, S.K.: Moment maps and diffeomorphisms. Asian J. Math. 3, 1–16 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Evans, L.C.: Classical solutions of fully nonlinear, convex, secondorder elliptic equations. Commun. Pure Appl. Math. 35, 333–363 (1982)CrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hou, Z.-L.: Complex Hessian equation on Kähler manifolds. Int. Math. Res. Not. 2009(16), 3098–3111 (2009)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Krylov, N.V.: Boundedly nonhomogeneous elliptic and parabolic equations. Izvestiya Ross. Akad. Nauk. SSSR 46, 487–523 (1982)zbMATHGoogle Scholar
  21. 21.
    Lejmi, M., Székelyhidi, G.: The J-flow and stability. Adv. Math. 274, 404–431 (2015)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sun, W.: Parabolic complex Monge–Ampère type equations on closed Hermtian manifolds. Calc. Var. PDE 54(4), 3715–3733 (2015)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Székelyhidi, G.: Fully non-linear elliptic equations on compact Hermitian manifolds. J. Differ. Geom. 109(2), 337–378 (2018)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tosatti, V., Weinkove, B.: The complex Monge–Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23, 1187–1195 (2010)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Trudinger, N.S.: Fully nonlinear, uniformly elliptic equations under natural structure conditions. Trans. Am. Math. Soc. 278(2), 751–769 (1983)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Weinkove, B.: Convergence of the J-flow on Kähler surfaces. Commun. Anal. Geom. 12, 949–965 (2004)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  40. 40.
    Zhang, D.-K.: Hessian equations on closed Hermitian manifolds. Pac. J. Math. 291(2), 485–510 (2017)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

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