Instantaneously complete Chern–Ricci flow and Kähler–Einstein metrics

  • Shaochuang Huang
  • Man-Chun LeeEmail author
  • Luen-Fai Tam


In this work, we obtain some existence results of Chern–Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as \(t\rightarrow 0\). These results can be viewed as a generalization of an existence result of Ricci flow by Giesen and Topping for surfaces of hyperbolic type to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete non-compact Hermitian manifolds, which generalizes the work of Lott–Zhang and Tosatti–Weinkove to complete non-compact Hermitian manifolds with possibly unbounded curvature.

Mathematics Subject Classification

Primary 32Q15 Secondary 53C44 



  1. 1.
    Aubin, T.: Équations du type Monge-Ampère sur les variétés kähleriennes compactes. C. R. Acad. Sci. Paris Sér. A-B, 283(3), A119–A121 (1976)Google Scholar
  2. 2.
    Boucksom, S., Guedj, V.: Regularizing properties of the Kähler–Ricci flow. In: An Introduction to the Kähler-Ricci Flow. Lecture Notes in Mathematics, vol. 2086, pp. 189–237. Springer, Cham (2013)Google Scholar
  3. 3.
    Cao, H.-D.: Deforming of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chau, A.: Convergence of the Kähler–Ricci flow on noncompact Kähler manifolds. J. Differ. Geom. 66(1), 211–232 (2004)CrossRefGoogle Scholar
  5. 5.
    Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82(2), 363–382 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cheng, S.-Y., Yau, S.-T.: On the existence of a complete Kähler Metric on noncompact complex manifolds and the regularity of Fefferman’s equation. Commun. Pure Appl. Math. 33(4), 507–544 (1980)CrossRefGoogle Scholar
  7. 7.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications: Part II: Analytic Aspects, Mathematical Surveys and Monographs, 144. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar
  8. 8.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications, Part III. Geometric-Analytic Aspects. Mathematical Surveys and Monographs, vol. 163. American Mathematical Society, Providence (2010)Google Scholar
  9. 9.
    Ge, H., Lin, A., Shen, L.-M.: The Kähler–Ricci flow on pseudoconvex domain. arXiv:1803.07761
  10. 10.
    Giesen, G., Topping, P.-M.: Ricci flow of negatively curved incomplete surfaces. Calc. Var. PDE 38, 357–367 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giesen, G., Topping, P.-M.: Existence of Ricci flows of incomplete surfaces. Commun. Partial Differ. Equ. 36, 1860–1880 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Giesen, G., Topping, P.-M.: Ricci flows with unbounded curvature. Math. Zeit. 273, 449–460 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gill, M.: Convergence of the parabolic complex Monge–Ampère equation on compact Hermitian manifolds. Commun. Anal. Geom. 19(2), 277–303 (2011)CrossRefGoogle Scholar
  14. 14.
    He, F., Lee, M.-C.: Weakly PIC1 manifolds with maximal volume growth. arXiv:1811.03318
  15. 15.
    Huang, S.-C.: A note on existence of exhaustion functions and its applications. J. Geom. Anal. 29(2), 1649–1659 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huang, S.-C., Lee, M.-C., Tam L.-F., Tong, F.: Longtime existence of Kähler–Ricci flow and holomorphic sectional curvature. arXiv:1805.12328
  17. 17.
    Lee, M.-C.; Tam, L.-F.: Chern-Ricci flows on noncompact manifolds. To appear in J. Differ. Geom. arXiv:1708.00141
  18. 18.
    Lott, J., Zhang, Z.: Ricci flow on quasi-projective manifolds. Duke Math. J. 156(1), 87–123 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ni, L., Tam, L.-F.: Poincarè–Lelong equation via the Hodge-Laplace heat equation. Compos. Math. 149(11), 1856–1870 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159
  21. 21.
    Sherman, M., Weinkove, B.: Interior derivative estimates for the Kähler–Ricci flow. Pac. J. Math. 257(2), 491–501 (2012)CrossRefGoogle Scholar
  22. 22.
    Sherman, M., Weinkove, B.: Local Calabi and curvature estimates for the Chern–Ricci flow. N. Y. J. Math. 19, 565–582 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Shi, W.-X.: Ricci deformation of the metric on complete Riemannian manifolds. J. Differ. Geom. 30, 303–394 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shi, W.-X.: Ricci flow and the uniformization on complete noncompact Kähler manifolds. J. Differ. Geom. 45, 94–220 (1997)CrossRefGoogle Scholar
  25. 25.
    Song, J., Tian, G.: The Kähler–Ricci flow through singularities. Invent. Math. 207(2), 519–595 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tam, L.-F.: Exhaustion functions on complete manifolds, recent advances in geometric analysis. Adv. Lect. Math. (ALM) 11, 211–215 (2010)zbMATHGoogle Scholar
  27. 27.
    Tian, G., Zhang, Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179192 (2006)Google Scholar
  28. 28.
    Tô, T.-D.: Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds. Math. Ann. 372(1–2), 699–741 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Topping, P.-M.: Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics. J. Eur. Math. Soc. (JEMS) 12, 1429–1451 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Tosatti, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern–Ricci form. J. Differ. Geom. 99(1), 125–163 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)CrossRefGoogle Scholar
  32. 32.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsNorthwestern UniversityEvanstonUSA
  3. 3.The Institute of Mathematical Sciences and Department of MathematicsThe Chinese University of Hong KongShatinChina

Personalised recommendations