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A Chern–Calabi Flow on Hermitian Manifolds

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Abstract

We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott–Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern–Ricci-flat metric in the \(\partial {\bar{\partial }}\)-class of the initial metric.

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References

  1. Angella, D., Calamai, S., Spotti, C.: On the Chern–Yamabe problem. Math. Res. Lett. 24(3), 645–677 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bedulli, L., Vezzoni, L.: A parabolic flow of balanced metrics. J. Reine Angew. Math. 723, 79–99 (2017)

    MathSciNet  MATH  Google Scholar 

  3. Bedulli, L., Vezzoni, L.: A scalar Calabi-type flow in Hermitian Geometry: Short-time existence and stability. Ann. Sc. Norm. Super. Pisa Cl. Sci. (2020) https://doi.org/10.2422/2036-2145.201802_012

  4. Berman, R.J., Darvas, T., Lu, C.H.: Convexity of the extended K-energy and the large time behavior of the weak Calabi flow. Geometry Topol. 21, 2945–2988 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bryant, R., Xu, F.: Laplacian flow for closed G2-structures: short time behavior. arXiv:1101.2004

  6. Calabi, E.: Extremal Kähler metrics in seminar on differential geometry. Ann. Math. Stud. 102, 259–290 (1982)

    Google Scholar 

  7. Calabi, E.: Extremal Kähler Metrics II. In: Differential Geometry and Complex Analysis, pp. 95–114, Springer-Verlag, New York (1985)

  8. Chang, S.-C.: The 2-dimensional Calabi flow. Nagoya Math. J. 181, 63–73 (2006)

    Article  MathSciNet  Google Scholar 

  9. Chen, X., He, W.Y.: On the Calabi flow. Am. J. Math. 130, 539–570 (2008)

    Article  MathSciNet  Google Scholar 

  10. Chen, X.X., Cheng, J.: On the constant scalar curvature Kähler metrics, apriori estimates. arXiv:1712.06697 (2017)

  11. Chen, X.X., Cheng, J.: On the constant scalar curvature Kähler metrics, existence results. arXiv:1801.00656 (2018)

  12. Chen, X.X., Sun, S.: Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics. Ann. Math. 180, 407–454 (2014)

    Article  MathSciNet  Google Scholar 

  13. Chruściel, P.T.: Semi-global existence and convergence of solutions of the Robinson–Trautman (2-dimensional Calabi) equation. Commun. Math. Phys. 137, 289–313 (1991)

    Article  MathSciNet  Google Scholar 

  14. Feng, R., Huang, H.: The global existence and convergence of the Calabi flow on \({{\mathbb{C}}}^n/{{\mathbb{Z}}}^n + i{{\mathbb{X}}}^n\). J. Funct. Anal. 263, 1129–1146 (2012)

    Article  MathSciNet  Google Scholar 

  15. Fine, J.: Calabi flow and projective embeddings. J. Differ. Geom. 84(3), 489–523 (2010)

    Article  MathSciNet  Google Scholar 

  16. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (1983)

    MATH  Google Scholar 

  17. Gill, M.: Convergence of the parabolic complex Monge–Ampère equation on compact Hermitian manifolds. Commun. Anal. Geom. 19(1), 65–78 (2011)

    MATH  Google Scholar 

  18. He, W.: On the convergence of the Calabi flow. Proc. Am. Math. Soc. 143, 1273–1281 (2015)

    Article  MathSciNet  Google Scholar 

  19. Huang, H., Zheng, K.: Stability of the Calabi flow near an extremal metric. Ann. Sc. Norm. Super. Pisa Cl. Sci. 11, 167–175 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Kawamura, M.: A scalar Calabi-type flow in the almost Hermitian geometry. Tsukuba J. Math. 43(1), 37–54 (2019)

    Article  MathSciNet  Google Scholar 

  21. Li, H., Wang, B., Zheng, K.: Regularity scales and convergence of the Calabi flow. J. Geom. Anal. 28, 2050–2101 (2018)

    Article  MathSciNet  Google Scholar 

  22. Phong, D.H., Picard, S., Zhang, X.: New curvature flows in complex geometry. Surv. Differ. Geometry 22, 331–364 (2017)

    Article  MathSciNet  Google Scholar 

  23. Picard, S.: Calabi+-Yau Manifolds with Torsion and Geometric Flows. CIME Summer School. http://people.math.harvard.edu/~spicard/cetraro.pdf (2018)

  24. Shen, X.S.: Estimates for metrics of constant Chern scalar curvature. arXiv: 1909.13445 (2019)

  25. Streets, J.: Long time existence of minimizing movement solutions of Calabi flow. Adv. Math. 259, 688–729 (2014)

    Article  MathSciNet  Google Scholar 

  26. Streets, J.: The consistency and convergence of K-energy minimizing movements. Trans. Am. Math. Soc. 368, 5075–5091 (2016)

    Article  MathSciNet  Google Scholar 

  27. Streets, J., Tian, G.: Hermitian curvature flow. J. Eur. Math. Soc. 13, 601–634 (2011)

    Article  MathSciNet  Google Scholar 

  28. Székelyhidi, G.: Remark on the Calabi flow with bounded curvature. Univ. Iagel. Acta Math. 50, 107–115 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Tian, G.: Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2000)

  30. Tosatti, V., Wang, Y., Weinkove, B., Yang, X.: \(C^{2,\alpha }\) estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc. Var. Partial Differ. Equ. 54(1), 431–453 (2015)

    Article  Google Scholar 

  31. Tosatti, V., Weinkove, B.: The Calabi flow with small initial energy. Math. Res. Lett. 40, 1033–1039 (2007)

    Article  MathSciNet  Google Scholar 

  32. Tosatti, V., Weinkove, B.: The complex Monge–Ampère equation on compact Hermitian manifolds. J. Am. Math. Soc. 23, 1187–1195 (2010)

    Article  Google Scholar 

  33. Tosatti, V., Weinkove, B.: The Chern–Ricci flow on complex surfaces. Comput. Math. 149, 2101–2138 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Tosatti, V., Weinkove, B.: On the evolution of a Hermitian metric by its Chern–Ricci form. J. Differ. Geom. 99(1), 125–163 (2015)

    Article  MathSciNet  Google Scholar 

  35. Ustinovskiy, Y.: Hermitian curvature flow on manifolds with non-negative Griffiths curvature. Am. J. Math. 141(6), 1751–1775 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is very grateful to her thesis advisor Ben Weinkove for his helpful suggestions and his continued support and encouragement. She would also like to thank Gregory Edwards, Antoine Song and Jonathan Zhu for some useful discussions, as well as the referee for many helpful constructive comments.

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  1. Columbia University, New York, NY, 10027, USA

    Xi Sisi Shen

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  1. Xi Sisi Shen
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Correspondence to Xi Sisi Shen.

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Shen, X.S. A Chern–Calabi Flow on Hermitian Manifolds. J Geom Anal 32, 129 (2022). https://doi.org/10.1007/s12220-021-00845-4

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