Mathematische Zeitschrift

, Volume 247, Issue 2, pp 279–302

Winding behaviour of finite-time singularities of the harmonic map heat flow*

  • Peter Topping


We settle a number of questions about the possible behaviour of the harmonic map heat flow at finite-time singularities. In particular, we show that a type of nonuniqueness of bubbles can occur at finite time, we show that the weak limit of the flow at the singular time can be discontinuous, we determine exactly the (polynomial) rate of blow-up in one particular example, and we show that ‘winding’ behaviour of the flow can lead to an unexpected failure of convergence when the flow is (locally) lifted to the universal cover of the target manifold.


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  1. 1.
    Bertsch, M., Dal Passo, R., Van Der Hout, R.: Nonuniqueness for the heat flow of harmonic maps on the disk. Arch. Ration. Mech. Anal. 161, 93–112 (2002)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Chang, K.-C.: Heat flow and boundary value problem for harmonic maps. Ann. Inst. Henri Poicaré, Analyse nonlinéaire 6 (5), 363–395 (1989)Google Scholar
  3. 3.
    Chang, K.-C., Ding, W.-Y., Ye, R.: Finite-time Blow-up of the Heat Flow of Harmonic Maps from Surfaces. J. Diff. Geom. 36, 507–515 (1992)MathSciNetMATHGoogle Scholar
  4. 4.
    Ding, W.-Y., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3, 543–554 (1995)MathSciNetMATHGoogle Scholar
  5. 5.
    Eells, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc. 10, 1–68 (1978)MathSciNetMATHGoogle Scholar
  6. 6.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–169 (1964)MATHGoogle Scholar
  7. 7.
    Freire, A.: Uniqueness for the harmonic map flow from surfaces to general targets. Comment. Math. Helv. 70, 310-338 (1995); correction in Comment. Math. Helv. 71, 330–337 (1996)MathSciNetMATHGoogle Scholar
  8. 8.
    Lemaire, L.: Applications harmoniques de surfaces riemannienes. J. Diff. Geom. 13, 51–78 (1978)MathSciNetMATHGoogle Scholar
  9. 9.
    Lin, F.-H., Wang, C.-Y.: Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. 6, 369–380 (1998)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Qing, J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Comm. Anal. Geom. 3, 297–315 (1995)MathSciNetMATHGoogle Scholar
  11. 11.
    Qing, J.: A remark on the finite time singularity of the heat flow for harmonic maps. Calc. Var. 17, 393–403 (2003)MATHGoogle Scholar
  12. 12.
    Qing, J., Tian, G.: Bubbling of the heat flows for harmonic maps from surfaces. Comm. Pure Appl. Math. 50, 295–310 (1997)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)MathSciNetMATHGoogle Scholar
  14. 14.
    Simon, L.: Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math. 118, 525–571 (1983)MathSciNetMATHGoogle Scholar
  15. 15.
    Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985)MathSciNetMATHGoogle Scholar
  16. 16.
    Topping, P.M.: Rigidity in the Harmonic Map Heat Flow. J. Diff. Geom. 45, 593–610 (1997)MathSciNetMATHGoogle Scholar
  17. 17.
    Topping, P.M.: An example of a nontrivial bubble tree in the harmonic map heat flow. Harmonic morphisms, harmonic maps and related topics (Brest, 1997) CRC Res. Notes Math. 413, 185–191 (2000)Google Scholar
  18. 18.
    Topping, P.M.: Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow. To appear, Annals of MathGoogle Scholar
  19. 19.
    Topping, P.M.: Reverse bubbling and nonuniqueness in the harmonic map flow. Int. Math. Res. Not. 10, 505–520 (2002)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Topping, P.M.: The Harmonic Map Heat Flow from Surfaces. PhD Thesis, University of Warwick (1996)Google Scholar
  21. 21.
    Wang, C.-Y.: Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. Houston J. Math. 22, 559–590 (1996)MathSciNetMATHGoogle Scholar
  22. 22.
    White, B.: Nonunique tangent maps at isolated singularities of harmonic maps. Bull. Am. Math. Soc. 26, 125–129 (1992)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Peter Topping
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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