Metric contraction of the cone divisor by the conical Kähler–Ricci Flow



We use the momentum construction of Calabi to study the conical Kähler–Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov–Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts only the cone divisor to a single point and Gromov–Hausdorff converges to a two dimensional projective orbifold. The limiting behaviour depends only on the cone angle, numerical properties of the initial Kähler class, and the degree of the Hirzebruch surface. This gives the first example of the conical Kähler–Ricci flow contracting the cone divisor to a single point, and shows that the conical flow may contract curves of self-intersection less than \((-\,1)\), as opposed to the smooth Kähler–Ricci flow. At the end, we introduce a conjectural picture of the geometry of finite time non-collapsing singularities of the flow on Kähler surfaces in general.



The results of this paper were contained in the author’s Ph.D. Thesis at Northwestern University [15].


  1. 1.
    Barth, W., Hulek, K., Peters, C., van de Ven, A.: Compact Complex Surfaces, Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, 2nd edn. Springer, Berlin (2004)Google Scholar
  2. 2.
    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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brendle, S.: Ricci flat Kähler metrics with edge singularities. Int. Math. Res. Not. 24, 5727–5766 (2013)CrossRefMATHGoogle Scholar
  4. 4.
    Calabi, E.: Extremal Kähler metrics. In: Seminar on Differential Geometry, Volume 102 of Annals of Mathematics Studies, pp. 259–290. Princeton University Press, Princeton (1982)Google Scholar
  5. 5.
    Campana, F., Guenancia, H., Păun, M.: Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields. Ann. Sc. Éc. Norm. Supér. (4) 46(6), 879–916 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cao, H.D.: Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359–372 (1985)MathSciNetCrossRefMATHGoogle Scholar
  7. 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)CrossRefMATHGoogle Scholar
  8. 8.
    Chen, X.X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds, II: limits with cone andgle less than \(2\pi \). J. Am. Math. Soc. 28(1), 199–234 (2015)CrossRefMATHGoogle Scholar
  9. 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)CrossRefMATHGoogle Scholar
  10. 10.
    Chen, X.X., Wang, Y.Q.: Bessel functions, heat kernel and the conical Kähler–Ricci flow. J. Funct. Anal. 269(2), 551–632 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, X.X., Wang, Y.Q.: On the long time behaviour of the conical Kähler–Ricci flows. arXiv:1402.6689 (2014)
  12. 12.
    Collins, T., Tosatti, V.: Kähler currents and null loci. Invent. Math. 202(3), 1167–1198 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Datar, V., Song, J.: A remark on Kähler metrics with conical singularities along a simple normal crossing divisor. Bull. Lond. Math. Soc. 47(6), 1010–1013 (2015)MathSciNetMATHGoogle Scholar
  14. 14.
    Donaldson, S.: Kähler metrics with cone singularities along a divisor. In: Essays in Mathematics and Its Applications, pp. 49–79. Springer, Heidelberg (2012)Google Scholar
  15. 15.
    Edwards, G.: The conical Kähler–Ricci flow and the analytic log minimal model program. PhD Thesis, Northwestern University (2018)Google Scholar
  16. 16.
    Edwards, G.: A scalar curvature bound along the conical Kähler–Ricci flow. J. Geom. Anal. 28(1), 225–252 (2018)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Eyssidieux, P., Guedj, V., Zeriahi, A.: Weak solutions to degenerate complex Monge–Ampère flows, II. Adv. Math. 293, 37–80 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons. J. Differ. Geom. 65(2), 169–209 (2003)CrossRefMATHGoogle Scholar
  19. 19.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, Oxford (1978)MATHGoogle Scholar
  20. 20.
    Guenancia, H., Păun, M.: Conic singularities metrics with prescribed Ricci curvature: general cone angles along normal crossing divisors. J. Differ. Geom. 103(1), 15–57 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Guo, B., Song, J.: Schauder estimates for equations with cone metrics, I. arXiv:1612.00075 (2016)
  22. 22.
    Guo, B., Song, J., Weinkove, B.: Geometric convergence of the Kähler–Ricci flow on surfaces of general type. Int. Math. Res. Not. 2016(18), 5652–5669 (2016)CrossRefGoogle Scholar
  23. 23.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hamilton, R.: Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5, 1–92 (1997)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jeffres, T.D., Mazzeo, R., Rubinstein, Y.A.: Kähler–Einstein metrics with edge singularities. Ann. of Math. (2) 183(1), 95–176 (2016)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Li, C., Sun, S.: Conic Kähler–Einstein metric revisited. Comm. Math. Phys. 331(3), 927–973 (2014)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Liu, J., Zhang, X.: The conical Kähler–Ricci flow on Fano manifolds. Adv. Math. 307, 1324–1371 (2017)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Liu, J., Zhang, X.: The conical Kähler–Ricci flow with weak initial data on Fano manifold. Int. Math. Res. Not. 17, 5343–5384 (2017)Google Scholar
  29. 29.
    Mazzeo, R., Rubinstein, Y.A., Sesum, N.: Ricci flow on surfaces with conic singularities. Anal. PDE 8(4), 839–882 (2015)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Nomura, R.: Blow-up behavior of the scalar curvature along the conical Kähler–Ricci flow with finite time singularities. Differ. Geom. Appl. 58, 1–16 (2018)CrossRefMATHGoogle Scholar
  31. 31.
    Perelman, G.: The entropy formula for Ricci flow and its geometric applications. arXiv:math/0211159 (2002)
  32. 32.
    Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109 (2003)
  33. 33.
    Phong, D.H., Song, J., Sturm, J., Wang, X.W.: The Ricci flow on the sphere with marked points. arXiv:1407.1118 (2014)
  34. 34.
    Phong, D.H., Song, J., Sturm, J., Wang, X.W.: Convergence of the conical Ricci flow on S2 to a soliton. arXiv:1503.04488 (2015)
  35. 35.
    Rubinstein, Y.: Smooth and singular Kähler–Einstein metrics. In: Geometric and Spectral Analysis, Volume 630 of Contemporary Mathematics, pp. 45–138. American Mathematical Society, Providence (2014)Google Scholar
  36. 36.
    Shen, L.: Maximal time existence of unnormalized conical Kähler–Ricci flow. arXiv:1411.7284 (2014)
  37. 37.
    Shen, L.: \(C^{2,\alpha }\)-estimate for conical Kähler–Ricci flow. Calc. Var. Partial Differ. Equ. 57(2), Art. 33, 15 (2018)CrossRefGoogle Scholar
  38. 38.
    Song, J., Tian, G.: The Kähler–Ricci flow through singularities. Invent. Math. 207(2), 519–595 (2017)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Song, J., Wang, X.: The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality. Geom. Topol. 20(1), 49–102 (2016)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Song, J., Weinkove, B.: The Kähler–Ricci flow on Hirzebruch surfaces. J. Reine Angew. 659, 141–168 (2011)MathSciNetMATHGoogle Scholar
  41. 41.
    Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler–Ricci flow. Duke Math. J. 162(2), 367–415 (2011)CrossRefMATHGoogle Scholar
  42. 42.
    Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler–Ricci flow, II. Proc. Lond. Math. Soc. 108(6), 1529–1561 (2014)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Tian, G.: Kähler–Einstein metrics on algebraic manifolds. In: Transcendental Methods in Algebraic Geometry (Cetraro, 1994), Volume 1646 of Lecture Notes in Mathematics, pp. 143–185. Springer, Berlin (1996)Google Scholar
  44. 44.
    Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)CrossRefMATHGoogle Scholar
  45. 45.
    Tian, G., Zhang, Z.: On the Kähler–Ricci flow on projective manifolds of general type. Chin. Ann. Math. 27(2), 179–192 (2006)CrossRefMATHGoogle Scholar
  46. 46.
    Tian, G., Zhang, Z.: Convergence of Kähler–Ricci flow on lower-dimensional algebraic manifolds of general type. Int. Math. Res. Not. 2016(21), 6493–6511 (2016)CrossRefGoogle Scholar
  47. 47.
    Troyanov, M.: Prescribing curvature on compact surfaces with conic singularities. Trans. Am. Math. Soc. 324, 793–821 (1991)CrossRefMATHGoogle Scholar
  48. 48.
    Tsuji, H.: Existence and degeneration of Kähler–Einstein metrics on minimal algebraic varieties of general type. Math. Ann. 281, 123–133 (1988)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Wang, B.: The local entropy along Ricci flow—Part A: the no local collapsing theorems. arXiv:1706.08485 (2017)
  50. 50.
    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)CrossRefMATHGoogle Scholar
  51. 51.
    Yin, H.: Ricci flow on surfaces with conical singularities. J. Geom. Anal. 20, 970–995 (2010)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Yin, H.: Ricci flow on surfaces with conical singularities, II. arXiv:1305.4355 (2013)
  53. 53.
    Zhang, Y.: A note on conical Kähler–Ricci flow on minimal elliptic Kähler surfaces. Acta Math. Sci. Ser. B Engl. Ed. 38(1), 169–176 (2018)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA

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