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Compactness of Kähler metrics with bounds on Ricci curvature and \({\mathcal {I}}\) functional

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Abstract

We prove a compactness theorem for Kähler metrics with various bounds on Ricci curvature and the \({\mathcal {I}}\) functional. We explore applications of our result to the continuity method and the Calabi flow.

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References

  1. Berman, R., Darvas, T., Lu, C.H.: Convexity of the extended K-energy and the long time behavior of the Calabi flow. Geom. Topol. 21(5), 2945–2988 (2017)

    Article  MathSciNet  Google Scholar 

  2. Berman, R., Darvas, T., Lu, C.H.: Regularity of weak minimizers of the K-energy and applications to properness and K-stability. arXiv:1602.03114

  3. Boucksom, S., Berman, R., Eyssidieux, P., Guedj, V., Zeriahi, A.: Kähler–Einstein metrics and the Kähler-Ricci flow on log Fano varieties. arXiv:1111.7158

  4. Chen, X.X.: The space of Kähler metrics. J. Differ. Geom. 56(2), 189–234 (2000)

    Article  Google Scholar 

  5. Chen, X.X.: On the existence of constant scalar curvature Kähler metric: a new perspective. arXiv:1506.06423

  6. Chen, X.X., He, W.: On the Calabi flow. Am. J. Math. 130(2), 539–570 (2008)

    Article  MathSciNet  Google Scholar 

  7. Chen, X.X., He, W.: The complex Monge–Ampère equation on compact Kähler manifolds. Math. Ann. 354(4), 1583–1600 (2012)

    Article  MathSciNet  Google Scholar 

  8. Darvas, T.: The Mabuchi geometry of finite energy classes. Adv. Math. 285, 182–219 (2015)

    Article  MathSciNet  Google Scholar 

  9. Darvas, T.: Geometric pluripotential theory on Kähler manifolds, preprint, available on author’s website (2017)

  10. Darvas, T., Rubinstein, Y.A.: Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics. J. Am. Math. Soc. 30(2), 347–387 (2017)

    Article  Google Scholar 

  11. Donaldson, S.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Eliashberg, Y., Fuchs, D., Ratiu, T., Weinstein, A. (eds.) Northern California Symplectic Geometry Seminar, vol. 196, p. 13. American Mathematical Society, Providence (1997)

    Google Scholar 

  12. Guedj, V., Zeriahi, A.: The weighted Monge–Ampère energy of quasiplurisubharmonic functions. J. Funct. Anal. 250(2), 442–482 (2007)

    Article  MathSciNet  Google Scholar 

  13. Fine, J.: Constant scalar curvature Kähler metrics on fibred complex surfaces. J. Differ. Geom. 68(3), 397–432 (2004)

    Article  Google Scholar 

  14. Hashimoto, Y.: Existence of twisted constant scalar curvature Kähler metrics with a large twist. arXiv:1508.00513 (2015)

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

    Article  MathSciNet  Google Scholar 

  16. He, W., Zeng, Y.: Constant scalar curvature equation and regularity of its weak solution. arXiv:1705.01236

  17. Li, H., Wang, B., Zheng, K.: Regularity scales and convergence of the Calabi flow. arXiv:1501.01851

  18. Kolodziej, S.: The Complex Monge–Ampere Equation and Pluripotential Theory, vol. 178. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  19. Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)

    Article  MathSciNet  Google Scholar 

  20. Song, J., Tian, G.: The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609–653 (2007)

    Article  MathSciNet  Google Scholar 

  21. Stoppa, J.: Twisted constant scalar curvature Kähler metrics and Kähler slope stability. J. Differ. Geom. 83(3), 663–691 (2009)

    Article  MathSciNet  Google Scholar 

  22. Szekelyhidi, G.: Remark on the Calabi flow with bounded curvature. Univ. Iagell. Acta Math. 50, 107–115 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Tian, G.: Canonical Metrics in Kahler Geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2000)

    Book  Google Scholar 

  24. Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation, I. Commun. Pure Appl. Math. 31, 339–411 (1978)

    Article  Google Scholar 

  25. Zeriahi, A.: Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions. Indiana Univ. Math. J. 50(1), 671–703 (2001)

    Article  MathSciNet  Google Scholar 

  26. Zeng, Y.: Deformations from a given Kähler metric to a twisted cscK metric. arXiv:1507.06287

Download references

Acknowledgements

The first named author has been partially supported by NSF grant DMS–1515795. The second named author has been partially supported by NSF grant DMS–1610202 and BSF grant 2012236. The third named author has been partially supported by NSF grant DMS–1611797.

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, USA

    Xiuxiong Chen

  2. University of Maryland, College Park, USA

    Tamás Darvas

  3. University of Oregon, Eugene, USA

    Weiyong He

Authors
  1. Xiuxiong Chen
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  2. Tamás Darvas
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  3. Weiyong He
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Correspondence to Tamás Darvas.

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Communicated by A. Neves.

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Chen, X., Darvas, T. & He, W. Compactness of Kähler metrics with bounds on Ricci curvature and \({\mathcal {I}}\) functional. Calc. Var. 58, 139 (2019). https://doi.org/10.1007/s00526-019-1572-6

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