Skip to main content
SpringerLink
Log in

Singularity formation in the Yang-Mills Flow

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract.

It is shown that, for the Yang-Mills flow, a sequence of blow-ups of a rapidly forming singularity will converge, modulo the gauge group, to a non-trivial homothetically shrinking soliton. Explicit examples of homothetically shrinking solitons are given in the case of trivial bundles over R n for \(5 \le n \le 9\).

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Chen, Y., Shen, C-L.: Monotonicity formula and small action regularity for Yang-Mills flows in higher dimensions. Calc. Var. 2, 389-403 (1994)

    MathSciNet  Google Scholar 

  2. Donaldson, S. K.: Anti-self-dual yang-mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3), 50, 1-26 (1985)

    Google Scholar 

  3. Donaldson, S. K., Kronheimer, P. B.: The geometry of four manifolds. Clarendon Press, Oxford 1990

  4. Gastel, A.: Singularities of first kind in the harmonic map and Yang-Mills heat flow. Math. Z. 242, 47-62 (2002)

    Article  Google Scholar 

  5. Grayson, M., Hamilton, R. S.: The formation of singularities in the harmonic map heat flow. Comm. Anal. and Geom. 4, 525-546 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Grotowski, J. F.: Finite time blow-up for the Yang-Mills heat flow in higher dimensions. Math. Zeit. 237, 321-333 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Hamilton, R. S.: Monotonocity formulas for parabolic flows on manifolds. Comm. Anal. and Geom. 1, 127-137 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Hörmander, L.: Linear partial differential operators. Springer, Berlin Heidelberg New York 1963

  9. Naito, H.: Finite time blowing-up for the Yang-Mills gradient flow in higher dimensions. Hokkaido Math. J. 23, 451-464 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Råde, J.: On the Yang-Mills heat equation in two and three dimensions. J. Reine Angew. Math. 431, 123-163 (1992)

    MathSciNet  Google Scholar 

  11. Schlatter, A., Struwe, M., Tavildah-Zadeh, A. S.: Global existence of the equivariant Yang-Mills heat flow in four space dimensions. Am. J. Math. 120, 117-128 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Struwe, M.: The Yang-Mills flow in four dimensions. Calc. Var. 2, 123-150 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Uhlenbeck, K. K.: Connections with L p bounds on curvature. Comm. Math. Phys. 83, 31-42 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ben Weinkove

Authors
  1. Ben Weinkove
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ben Weinkove.

Additional information

Received: 22 November 2002, Accepted: 12 March 2003, Published online: 6 June 2003

Mathematics Subject Classification (2000):

53C44

Rights and permissions

Reprints and permissions

About this article

Cite this article

Weinkove, B. Singularity formation in the Yang-Mills Flow. Cal Var 19, 211–220 (2004). https://doi.org/10.1007/s00526-003-0217-x

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-003-0217-x

Keywords

Search

Navigation