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A Scalar Curvature Bound Along the Conical Kähler–Ricci Flow

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Abstract

Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler–Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.

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References

  1. Aubin, T.: Équation du type Monge-Ampère sur les variétés Kählerienes compactes. C. R. Acad. Sci. Paris Sér. A-B 283, A119–A121 (1976)

    Google Scholar 

  2. Berman, R.: A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics. Adv. Math. 248, 1254–1297 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. 24, 5727–5766 (2013)

    Article  MATH  Google Scholar 

  5. Campana, F., Guenancia, H., Paun, M.: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Ann. Sci. Ec. Norm. Sup. 46, 879–916 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, H.D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X.X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities. J. Am. Math. Soc. 28(1), 183–197 (2015)

    Article  MATH  Google Scholar 

  8. Chen, X.X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds, II: limits with cone angle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)

    Article  MATH  Google Scholar 

  9. Chen, X.X., Donaldson, S., Sun, S.: Kähler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Am. Math. Soc. 28(1), 235–278 (2015)

    Article  MATH  Google Scholar 

  10. Chen, X.X., Wang, Y.Q.: Bessel functions, heat kernel and the conical Kähler-Ricci flow. arXiv:1305.0255

  11. Chen, X.X., Wang, Y.Q.: On the long time behaviour of the conical Kähler-Ricci flows. arXiv:1402.6689

  12. Cheng, S.Y., Yau, S.-T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Collins, T., Székelyhidi, G.: The twisted Kähler-Ricci flow. arXiv:1207.5441v2

  14. Collins, T., Tosatti, V.: Kähler currents and null loci. arXiv:1304.5216

  15. Demailly, J.-P., Pali, N.: Degenerate complex Monge-Ampère equations over compact Kähler manifolds. Int. J. Math. 21(3), 357–405 (2010)

    Article  MATH  Google Scholar 

  16. Donaldson, S.: Kähler metric with cone singularities along a divisor. arXiv:1102.1196

  17. Eyssidieux, P., Guedj, V., Zeriahi, A.: A priori \(L^\infty \)-estimates for degenerate complex Monge-Ampère equations. Int. Math. Res. Not, IMRN (2008)

  18. Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc 22, 607–639 (2009)

    Article  MATH  Google Scholar 

  19. Fong, F.T.-H., Zhang, Z.: The collapsing rate of the Kähler-Ricci flow with regular infinite time singularity. arXiv:1202.3199, to appear in J. Reine Angew. Math

  20. Gill, M.: Collapsing of products along the Kähler-Ricci flow. Trans. Am. Math. Soc. 366(7), 3907–3924 (2014)

    Article  MATH  Google Scholar 

  21. Gross, M., Tosatti, V., Zhang, Y.: Collapsing of abelian fibered Calabi-Yau manifolds. Duke Math. J. 162(3), 517–551 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guenancia, H., Paun, M.: Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors. arXiv:1307.6375

  23. Hacon, C., McKernan, J.: On the existence of flips. math.ucsd.edu/~jmckerna/Papers/existence.pdf

  24. Jeffres, T.D., Mazzeo, R., Rubinstein, Y.A.: Kähler-Einstein metrics with edge singularities, with an appendix by C. Li and Y.A. Rubinstein. arXiv:1105.5216

  25. Li, C., Sun, S.: Conic Kähler-Einstein metric revisited. Commun. Math. Phys. 331(3), 927–973 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu, J., Zhang, X.: The conical Kähler-Ricci flow on Fano manifolds. arXiv:1402.1832

  27. Mazzeo, R., Rubinstein, Y.A., Sesum, N.: Ricci flow on surfaces with conic singularities. arXiv:1306.6688

  28. Perelman, G.: Unpublished work on the Kähler-Ricci flow

  29. Phong, D.H., Song, J., Sturm, J., Wang, X.W.: Convergence of the conical Ricci flow on S2 to a soliton. arXiv:1503.04488

  30. Phong, D.H., Song, J., Sturm, J., Wang, X.W.: The Ricci flow on the sphere with marked points. arXiv:1407.1118

  31. Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Kähler-Ricci flow and the \({\bar{\partial }}\) operator on vector fields. J. Differ. Geom. 81(3), 631–647 (2009)

    Article  MATH  Google Scholar 

  32. Phong, D.H., Sturm, J.: On stability and the convergence of the Kähler-Ricci flow. J. Differ. Geom. 72(1), 149–168 (2006)

    Article  MATH  Google Scholar 

  33. Rubinstein, Y.: Smooth and singular Kähler-Einstein metrics. arXiv:1404.7451

  34. Sesum, N., Tian, G.: Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Jussieu 7(3), 575–587 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shen, L.: Unnormalize conical Kähler-Ricci flow. arXiv:1411.7284

  36. Song, J., Tian, G.: Bounding scalar curvature for global solutions of the Kähler-Ricci flow. arXiv:1111.5681

  37. Song, J., Tian, G.: The Kähler-Ricci flow through singularities. arXiv:0909.4898

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Song, J., Tian, G.: Canonical measures and Kähler-Ricci flow. J. Am. Math. Soc 25, 303–353 (2012)

    Article  MATH  Google Scholar 

  40. Song, J., Wang, X.W.: The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. arXiv:1207.4839

  41. Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler-Ricci flow. Duke Math. J. 162(2), 367–415 (2011)

    Article  MATH  Google Scholar 

  42. Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler-Ricci flow, II. Proc. Lond. Math. Soc. 108(6), 1529–1561 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tian, G., Zhang, Z.: Regularity of Kähler-Ricci flows on Fano manifolds. arXiv:1310.5897

  44. Tian, G., Zhang, Z.: On the Kähler-Ricci flow on projective manifolds of general type. Chin. Ann. Math. 27(2), 179 (2006)

    Article  MATH  Google Scholar 

  45. Tian, G., Zhu, X.H.: Convergence of the Kähler-Ricci flow. J. Am. Math. Soc. 20(3), 675–699 (2007)

    Article  MATH  Google Scholar 

  46. Tosatti, V., Weinkove, B., Yang, X.: Käher-Ricci flow, Ricci-flat metrics and collapsing limits. arXiv:1408.0161

  47. Tsuji, H.: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281, 123–133 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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)

    Article  MATH  Google Scholar 

  49. Zhang, Z.: Scalar curvature bound for Kähler-Ricci flows over minimal manifolds of general type. Int. Math. Res. Not. (2009). doi:10.1093/imrn/rnp073

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Acknowledgements

This article would not exist were it not for the continued support and counsel of my thesis advisors Valentino Tosatti and Ben Weinkove. The author also thanks them for their many helpful conversations and improvements toward the final version of this paper.

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  1. Department of Mathematics, Northwestern University, Evanston, IL, USA

    Gregory Edwards

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  1. Gregory Edwards
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Edwards, G. A Scalar Curvature Bound Along the Conical Kähler–Ricci Flow. J Geom Anal 28, 225–252 (2018). https://doi.org/10.1007/s12220-017-9817-0

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  • DOI: https://doi.org/10.1007/s12220-017-9817-0

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