Communications in Mathematics and Statistics

, Volume 5, Issue 3, pp 277–316 | Cite as

The Flow of Gauge Transformations on Riemannian Surface with Boundary

  • Wanjun Ai


We consider the gauge transformations of a metric G-bundle over a compact Riemannian surface with boundary. By employing the heat flow method, the local existence and the long time existence of generalized solution are proved.


Heat flow Coulomb gauge Blow-up analysis 

Mathematics Subject Classification

58J35 58E15 


  1. 1.
    Chang, K.-C.: Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(5), 363–395 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs. The Clarendon Press, New York (1990)MATHGoogle Scholar
  3. 3.
    Eells Jr., J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Freed, D.S., Uhlenbeck, K.K.: Instantons and Four-Manifolds, Second, Mathematical Sciences Research Institute Publications, vol. 1. Springer-Verlag, New York (1991)Google Scholar
  5. 5.
    Fröhlich, S., Müller, F.: On the existence of normal Coulomb frames for two-dimensional immersions with higher codimension. Analysis (Munich) 31(3), 221–236 (2011)MathSciNetMATHGoogle Scholar
  6. 6.
    Hamilton, R.S.: Harmonic Maps of Manifolds with Boundary. Lecture Notes in Mathematics, vol. 471. Springer-Verlag, Berlin (1975)CrossRefGoogle Scholar
  7. 7.
    Hélein, F.: Harmonic Maps, Conservation Laws and Moving Frames. Second, Cambridge Tracts in Mathematics, vol. 150. Cambridge University Press, Cambridge (2002). Translated from the 1996 French original, With a foreword by James EellsGoogle Scholar
  8. 8.
    Jost, J.: Riemannian Geometry and Geometric Analysis. Sixth Universitext. Springer, Heidelberg (2011)CrossRefMATHGoogle Scholar
  9. 9.
    LadyŽenskaja, O.A., Solonnikov, V.A., Ural0ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)Google Scholar
  10. 10.
    Lawson Jr., H.B.: The Theory of Gauge Fields in Four Dimensions. CBMS Regional Conference Series in Mathematics, vol. 58. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence (1985)Google Scholar
  11. 11.
    Li, Y., Wang, Y.: Bubbling location for F-harmonic maps and inhomogeneous Landau–Lifshitz equations. Comment. Math. Helv. 81(2), 433–448 (2006)MathSciNetMATHGoogle Scholar
  12. 12.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co., Inc, River Edge (1996)CrossRefMATHGoogle Scholar
  13. 13.
    Ma, L.: Harmonic map heat flow with free boundary. Comment. Math. Helv. 66(2), 279–301 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    Meyer, Y., Rivière, T.: A partial regularity result for a class of stationary Yang–Mills fields in high dimension. Rev. Mat. Iberoam. 19(1), 195–219 (2003)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. American Mathematical Society, Providence (2008). Accessed 15 May 2017
  16. 16.
    Müller, F., Schikorra, A.: Boundary regularity via Uhlenbeck–Rivi‘ere decomposition. Analysis (Munich) 29(2), 199–220 (2009)MathSciNetMATHGoogle Scholar
  17. 17.
    Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3) 20, 733–737 (1966)MathSciNetMATHGoogle Scholar
  18. 18.
    Qing, J.: On singularities of the heat flow for harmonic maps from surfaces into spheres. Commun. Anal. Geom. 3(1–2), 297–315 (1995)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Qing, J.: A remark on the finite time singularity of the heat flow for harmonic maps. Calc. Var. Partial Differ. Equ. 17(4), 393–403 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Rivière, T., Struwe, M.: Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math. 61(4), 451–463 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. (2) 113(1), 1–24 (1981)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Schikorra, A.: A remark on gauge transformations and the moving frame method. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2), 503–515 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Schoen, R.M.: Analytic aspects of the harmonic map problem. In: Chern, S.S. (ed.) Seminar on Nonlinear Partial Differential Equations, vol. 2, pp. 321–358. Springer, New York (1984). doi: 10.1007/978-1-4612-1110-5_17 CrossRefGoogle Scholar
  26. 26.
    Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60(4), 558–581 (1985)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tao, T., Tian, G.: A singularity removal theorem for Yang–Mills fields in higher dimensions. J. Am. Math. Soc. 17(3), 557–593 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Topping, P.: Winding behaviour of finite-time singularities of the harmonic map heat flow. Math. Z. 247(2), 279–302 (2004)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Uhlenbeck, K.K.: Connections with Lpbounds on curvature. Commun. Math. Phys. 83(1), 31–42 (1982)CrossRefMATHGoogle Scholar
  30. 30.
    Wang, C.: Biharmonic maps from R4 into a Riemannian manifold. Math. Z. 247(1), 65–87 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wang, C.: Stationary biharmonic maps from Rm into a Riemannian manifold. Commun. Pure Appl. Math. 57(4), 419–444 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

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