Skip to main content
SpringerLink
Log in

Parabolic flows in complex geometry

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

An informal introduction is provided to some current problems on canonical metrics in complex geometry, from the point of view of non-linear partial differential equations. The emphasis is on the parabolic flow approach and open problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, T.: Equations du type Monge–Ampère sur les variétés Kähleriennes compactes. Bull. Sci. Math. 102, 63–95 (1978)

    MATH  MathSciNet  Google Scholar 

  2. Aubin, T.: Meilleures constantes dans le theoreme d’inclusion de Sobolev et un theoreme de Fredholm non-lineaires pour la transformation conforme de la courbure scalaire. J. Funct. Anal. 32, 148–174 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bando, S., Siu, Y.T.: Stable sheaves and Einstein-Hermitian metrics. In: Geometry and Analysis on Complex Manifolds, pp. 39–50. World Scientific, River Edge (1994)

    Chapter  Google Scholar 

  4. Belavin, A., Polyakov, A., Schwarz, A., Tyuptin, Y.: Pseudoparticle solutions of the Yang–Mills equations. Phys. Lett. B 59, 85–88 (1975)

    Article  MathSciNet  Google Scholar 

  5. Berndtsson, B.: A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (20 Mar 2013, submitted). arXiv:1303.4975

  6. Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46–74 (1985)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, X.X.: Calabi flow in Riemann surfaces revisited: a new point of view. Int. Math. Res. Not. 2001(6), 275–297 (2001)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, X.X., He, W.: The Calabi flow on toric Fano surfaces. Math. Res. Lett. 17(2), 231–241 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, X.X., Donaldson, S., Sun, S.: Kähler–Einstein metrics and stability (19 Nov 2012, submitted). arXiv:1210.7494

  12. Chen, X.X., Donaldson, S., Sun, S.: Kähler–Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities. arXiv:1211.4566

  13. Chen, X.X., Donaldson, Sun S, S.: Kähler–Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π (19 Dec 2012, submitted). arXiv:1212.4714

  14. Chen, X.X., Donaldson, Sun S, S.: Kähler–Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2 and completion of the main proof (1 Feb 2013, submitted). arXiv:1302.0282

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Collins, T., Jacob, A.: On the bubbling set of the Yang–Mills flow on a compact Kähler manifold (28 Jun 2012 (v1), submitted; 30 Sep 2013 (this version, v3), last revised). arXiv:1206.6790

  17. Collins, T., Szekelyhidi, G.: The twisted Kähler–Ricci flow. (23 Jul 2012 (v1), submitted; 6 Nov 2012 (this version, v2), last revised). arXiv:1207.5441

  18. Daskalopoulos, G., Wentworth, R.: Convergence properties of the Yang–Mills flow on Kähler surfaces. J. Reine Angew. Math. 575, 69–99 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Daskalopoulos, G., Wentworth, R.: On the blow-up set of the Yang–Mills flow on Kähler surfaces. Math. Z. 256(2), 301–310 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Donaldson, S.K.: Anti self dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  21. Donaldson, S.K.: Infinite determinants, stable bundles, and curvature. Duke Math. J. 54, 231–247 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  22. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 59, 289–349 (2002)

    MathSciNet  Google Scholar 

  23. Donaldson, S.K.: Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal. 19(1), 83–136 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Donaldson, S.K., Kronheimer, P.: The Geometry of Four-Manifolds. Oxford University Press, London (1990)

    MATH  Google Scholar 

  25. Donaldson, S.K., Sun, S.: Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry (12 Jun 2012, submitted). arXiv:1206.2609

  26. Eells, J., Sampson, G.: Harmonic mappings into Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  27. Hamilton, R.S.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

    MATH  Google Scholar 

  28. Hamilton, R.S.: Formation of singularities in the Ricci flow. Surv. Differ. Geom. II, 7–136 (1995)

    Google Scholar 

  29. Hong, M.C., Tian, G.: Asymptotical behavior of the Yang–Mills flow and singular Yang–Mills connections. Math. Ann. 330(3), 441–472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ilmanen, T., Huisken, G.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)

    MATH  MathSciNet  Google Scholar 

  31. Jacob, A.: The limit of the Yang–Mills flow on semi-stable bundles (25 Apr 2011 (v1), submitted; 26 Aug 2013 (this version, v2), last revised). arXiv:1104.4767

  32. Jacob, A.: The Yang–Mills flow and the Atiyah–Bott formula for compact Kähler manifolds (7 Sep 2011 (v1), submitted; 1 May 2012 (this version, v2), last revised). arXiv:1109.1550

  33. Kolodziej, S.: The complex Monge–Ampère equation. Acta Math. 180, 69–117 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Li, Q., Xu, C.: Special test configurations and K-stability of Fano varieties (23 Nov 2011 (v1), submitted; 14 Oct 2013 (this version, v3), last revised). arXiv:1111.5398

  35. Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (11 Nov 2002, submitted). arXiv:math.DG/0211159

  36. Perelman, G.: Unpublished announcement

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

    MATH  MathSciNet  Google Scholar 

  38. Phong, D.H., Sturm, J.: Lectures on stability and constant scalar curvature. In: Handbook of Geometric Analysis Vol. III, ALM 14. Higher Education Press/International Press, Beijing/Boston (2009)

    Google Scholar 

  39. Phong, D.H., Sesum, N., Sturm, J.: Multiplier ideal sheaves and the Kähler–Ricci flow. Commun. Anal. Geom. 15(3), 613–632 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  40. Phong, D.H., Song, J., Sturm, J., Weinkove, B.: The Moser–Trudinger inequality on Kähler–Einstein manifolds. Am. J. Math. 130(4), 1067–1085 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. 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)

    MATH  MathSciNet  Google Scholar 

  42. Phong, D.H., Song, J., Sturm, J., Weinkove, B.: On the convergence of the modified Kähler–Ricci flow and solitons. Comment. Math. Helv. 86(1), 91–112 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  43. Sibley, B.: Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit (24 Jun 2012 (v1), submitted; 2 Jul 2013 (this version, v3), last revised). arXiv:1206.5491

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

    Article  MATH  MathSciNet  Google Scholar 

  45. Song, J., Tian, G.: Canonical measures and Kähler–Ricci flow. J. Am. Math. Soc. 25(2), 303–353 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  46. Song, J., Weinkove, B.: The Kähler–Ricci flow on Hirzebruch surfaces. J. Reine Angew. Math. 659, 141–168 (2011)

    MATH  MathSciNet  Google Scholar 

  47. Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler–Ricci flow. Duke Math. J. 162(2), 367–415 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  48. Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler–Ricci flow II (9 Feb 2011 (v1), submitted; 20 Jul 2012 (this version, v2), last revised). arXiv:1102.1759

  49. Song, J., Weinkove, B.: Lecture notes on the Kähler–Ricci flow (15 Dec 2012, submitted). arXiv:1212.3653

  50. Stoppa, J.: K-stability of constant scalar curvature Kähler manifolds. Adv. Math. 221(4), 1397–1408 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  51. Struwe, M.: Curvature flows on surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1(2), 247–274 (2002)

    MATH  MathSciNet  Google Scholar 

  52. Szekelyhidi, G.: Optimal test-configurations for toric varieties. J. Differ. Geom. 80, 501–523 (2008)

    MATH  MathSciNet  Google Scholar 

  53. Szekelyhidi, G.: The Calabi functional on a ruled surface. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 837–856 (2009)

    MATH  MathSciNet  Google Scholar 

  54. Szekelyhidi, G.: Remark on the Calabi flow with bounded curvature (12 Sep 2012, submitted). arXiv:1209.2649

  55. Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  56. Tian, G.: K-stability and Kähler–Einstein metrics (20 Nov 2012 (v1), submitted; 28 Jan 2013 (this version, v2), last revised). arXiv:1211.4669

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

    Article  MATH  MathSciNet  Google Scholar 

  58. Tian, G., Zhu, X.H.: Convergence of the Kähler–Ricci flow on Fano manifolds, II (23 Feb 2011, submitted). arXiv:1102.4798

  59. Tosatti, V.: Kähler–Ricci flow on stable Fano manifolds. J. Reine Angew. Math. 640, 67–84 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  60. Tosatti, V., Weinkove, B.: The Calabi flow with small initial energy. Math. Res. Lett. 14(6), 1033–1039 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  61. Uhlenbeck, K.: Removable singularities in Yang–Mills fields. Commun. Math. Phys. 83, 11–29 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  62. Uhlenbeck, K., Yau, S.T.: On the existence of Hermitian Yang–Mills connections on stable vector bundles. Commun. Pure Appl. Math. 39, 257–293 (1986)

    Article  MathSciNet  Google Scholar 

  63. Wang, X.J., Zhu, X.H.: Kähler–Ricci solitons on toric manifolds with positive first Chern class. Adv. Math. 188, 87–103 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  64. 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 

  65. Yau, S.T.: Open problems in geometry. Proc. Symp. Pure Math. 54, 1–28 (1993)

    Article  Google Scholar 

  66. Zhang, Q.: A uniform Sobolev inequality under Ricci flow. Int. Math. Res. Not. 2007(17), rnm056 (2007). 17 pp.

    Google Scholar 

  67. Zhang, Z.: Kähler–Ricci flow on Fano manifolds with vanished Futaki invariants. Math. Res. Lett. 18, 969–982 (2011). arXiv:1010.5959

    Article  MATH  MathSciNet  Google Scholar 

  68. Zhang, Q.: Bounds on volume growth of geodesic balls under Ricci flow. Math. Res. Lett. 19, 245–253 (2012). arXiv:1107.4262

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to really thank Professor Ngo Bao Chau for his invitation to visit the Vietnam Institute for Advanced Study in Mathematics. He is extremely grateful to Professor Chau and his wonderful family for their warm hospitality, and for awakening in him deeply seated memories about Vietnam. He would also like to thank the many colleagues whom he finally had a chance to meet in Hanoi, including Professor Le Tuan Hoa (the Managing Director of the Institute), Professor Do Duc Thai, Professor Le Hung Son, and Professor Nguyen Minh Tuan. They all helped greatly in making his visit a truly memorable experience. Finally, he would like to express his great appreciation to the anonymous referee for a particularly careful reading of the paper.

Work supported in part by the National Science Foundation under Grant DMS-12-66033. Contribution to the Annual Meeting of the Vietnam Institute for Advanced Study in Mathematics, 2013.

Author information

Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Duong H. Phong

Authors
  1. Duong H. Phong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Duong H. Phong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Phong, D.H. Parabolic flows in complex geometry. Acta Math Vietnam. 39, 35–54 (2014). https://doi.org/10.1007/s40306-014-0046-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-014-0046-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation