Skip to main content
SpringerLink
Log in

A Note on Curvature Estimate of the Hermitian–Yang–Mills Flow

Dedicated to celebrate the Sixtieth anniversary of USTC

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

In this paper, we study the curvature estimate of the Hermitian–Yang–Mills flow on holomorphic vector bundles. In one simple case, we show that the curvature of the evolved Hermitian metric is uniformly bounded away from the analytic subvariety determined by the Harder–Narasimhan–Seshadri filtration of the holomorphic vector bundle.

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.

Similar content being viewed by others

References

  1. Alvarez-Consul, L., Garcis-Prada, O.: Dimensional reduction, SL(2, C)-equivariant bundles and stable holomorphic chains. Int. J. Math. 2, 159–201 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A 308, 524–615 (1982)

    MATH  Google Scholar 

  3. Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic geometry, Sendai, 1985, 11C40. Advanced Studies in Pure Mathematics, 10. North-Holland, Amsterdam (1987)

  4. Bando, S., Siu, Y.T.: Stable sheaves and Einstein-Hermitian metrics. In: Mabuchi, T., Noguchi, J., Ochiai, T. (eds.) Geometry and Analysis on Complex Manifolds, pp. 39–50. World Scientific, Singapore (1994)

    Chapter  Google Scholar 

  5. Bartolomeis, P.D., Tian, G.: Stability of complex vector bundles. J. Differ. Geom. 43, 232–275 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Biquard, O.: On parabolic bundles over a complex surface. J. Lond. Math. Soc 53(2), 302–316 (1996)

    Article  MATH  Google Scholar 

  7. Bradlow, S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990)

    Article  MATH  Google Scholar 

  8. Bradlow, S.B., Garcia-Prada, O.: Stable triples, equivariant bundles and dimensional reduction. Math. Ann. 304, 225–252 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cheng, S.Y., Li, P.: Heat kernel estimates and lower bound of eigenvalues. Comment. Math. Helv. 56, 327–338 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Collins, T., Jacob, A.: Remarks on the Yang–Mills flow on a compact Kähler manifold. Univ. Iagel. Acta Math. 51, 17–43 (2013)

    MATH  Google Scholar 

  11. Daskalopoulos, G.: The topology of the space of stable bundles on a Riemann surface. J. Differ. Geom. 36, 699–746 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Donaldson, S.K.: A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom. 18, 279–315 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  Google Scholar 

  16. Garcia-Prada, O.: Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math. 5, 1–52 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  17. Griffiths, P., Harris, J.: Principles of Algebraic Geometry, Pure and Applied Mathematics. Wiley-Interscience, New York (1978)

    MATH  Google Scholar 

  18. Grigor’yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Differ. Geom. 45(1), 33–52 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79(1), 109–203 (1964)

    Article  MATH  Google Scholar 

  20. Hironaka, H.: Flattening theorem in complex–analytic geometry. Am. J. Math. 97(2), 503–547 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Huybrechts, D., Lehn, M.: Stable pairs on curves and surfaces. J. Algebr. Geom. 4(1), 67–104 (1995)

    MathSciNet  MATH  Google Scholar 

  24. Jacob, A.: The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds. Amer. J. Math. 138(2), 329–365 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jost, J., Zuo, K.: Harmonic maps and \(Sl(r,C)\)-representations of fundamental groups of quasiprojective manifolds. J. Algebr. Geom. 5(1), 77–106 (1996)

    MathSciNet  MATH  Google Scholar 

  26. Li, J., Yau, S.T.: Hermitian–Yang–Mills connection on non-Kähler manifolds, Mathematical aspects of string theory (San Diego, Calif., 1986), pp. 560–573, Adv. Ser. Math. Phys., 1. World Scientific Publishing, Singapore (1987)

  27. Li, J.Y., Zhang, C.J., Zhang, X.: The limit of the Hermitian-Yang-Mills flow on reflexive sheaves. Adv. Math. 325, 165–214 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, P.: Geometry analysis. Cambridge Studies in Advanced Mathematics (No. 134)

  29. Li, P.: On the Sobolev constant and the \(p\)-spectrum of a compact Riemannian manifold. Ann. Sci. Ecole Norm. Sup. 13(4), 451–468 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, J.Y.: Hermitian–Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds. Comm. Anal. Geom 8(3), 445–475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, J.Y., Narasimhan, M.S.: Hermitian–Einstein metrics on parabolic stable bundles. Acta Math. Sin. (Engl. Ser.) 15(1), 93–114 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, P., Tam, L.F.: The heat equation and harmonic maps of complete manifold. Invent. Math. 105, 1–46 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J.Y., Zhang, X.: The gradient flow of Higgs pairs. J. Eur. Math. Soc. 13, 1373–1422 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, J.Y., Zhang, X.: Existence of approximate Hermitian–Einstein structures on semi-stable Higgs bundles. Calc. Var. Partial Differ. Equ. 52(3–4), 783–795 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Li, J., Zhang, X.: The limit of the Yang–Mills–Higgs flow on Higgs bundles. Int. Math. Res. Not. 2017(1), 232–276 (2017)

    Article  MathSciNet  Google Scholar 

  36. Li, J.Y., Zhang, C.J., Zhang, X.: Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var. Partial Differ. Equ. 56(3), 81, 33 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Mochizuki, T.: Kobayashi–Hitchin correspondence for tame harmonic bundles and an application. Astérisque 309 (2006). ISBN: 978-2-85629-226-6

  38. Mochizuki, T.: Kobayashi–Hitchin correspondence for tame harmonic bundles. Geom. Topol. 13(1), 359–455 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Mundet i Riera, I.: A Hitchin–Kobayashi correspondence for Kähler fibrations. J. Reine Angew. Math. 528, 41–80 (2000)

    MathSciNet  MATH  Google Scholar 

  40. Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector bundles on compact Riemann surfaces. Ann. Math. 82, 540–567 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sibley, B.: Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: the canonical structure of the limit. J. Reine Angew. Math. 706, 123–191 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Sibley, B., Wentworth, R.: Analytic cycles, Bott–Chern forms, and singular sets for the Yang–Mills flow on Kähler manifolds. Adv. Math. 279, 501–531 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Simpson, C.T.: Constructing variations of Hodge structures using Yang–Mills connections and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)

    Article  MATH  Google Scholar 

  44. Siu, Y.T.: Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics. Birkhauser, Basel-Boston, MR 89d, 32020 (1987)

  45. Uhlenbeck, K.K.: A priori estimates for Yang–Mills fields, unpublished manuscript

  46. Uhlenbeck, K.K.: Connections with \(L^p\) bounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)

    Article  MATH  Google Scholar 

  47. Uhlenbeck, K.K., Yau, S.T.: On existence of Hermitian–Yang–Mills connection in stable vector bundles. Comm. Pure Appl. Math. 39S, 257–293 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  48. Voisin, C.: Hodge Theory and Complex Algebraic Geometry. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  49. Wilkin, G.: Morse theory for the space of Higgs bundles. Comm. Anal. Geom. 16(2), 283–332 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Jiayu Li, Chuanjing Zhang & Xi Zhang

  2. AMSS, CAS, Beijing, 100080, People’s Republic of China

    Jiayu Li

Authors
  1. Jiayu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chuanjing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xi Zhang.

Additional information

The authors were supported in part by NSF in China, Nos. 11625106, 11571332, and 11721101.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, J., Zhang, C. & Zhang, X. A Note on Curvature Estimate of the Hermitian–Yang–Mills Flow. Commun. Math. Stat. 6, 319–358 (2018). https://doi.org/10.1007/s40304-018-0135-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-018-0135-z

Keywords

Mathematics Subject Classification

Search

Navigation