The Flow of Gauge Transformations on Riemannian Surface with Boundary

Abstract

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.

Keywords

Heat flow / Coulomb gauge / Blow-up analysis

References

[1] Chang K-C. Heat flow and boundary value problem for harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire. 1989, 6 5 363-395

[2] Donaldson SK, Kronheimer PB. The Geometry of Four-Manifolds. Oxford Mathematical Monographs. 1990 New York: The Clarendon Press

[3] Eells J Jr, Sampson JH. Harmonic mappings of Riemannian manifolds. Am. J. Math.. 1964, 86 109-160

[4] Freed DS, Uhlenbeck KK. Instantons and Four-Manifolds, Second, Mathematical Sciences Research Institute Publications. 1991 New York: Springer-Verlag

[5] Fröhlich S, Müller F. On the existence of normal Coulomb frames for two-dimensional immersions with higher codimension. Analysis (Munich). 2011, 31 3 221-236

[6] Hamilton RS. Harmonic Maps of Manifolds with Boundary. Lecture Notes in Mathematics. 1975 Berlin: Springer-Verlag

[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 Eells

[8] Jost J. Riemannian Geometry and Geometric Analysis. 2011 Heidelberg: Springer

[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)

[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)

[11] Li Y, Wang Y. Bubbling location for F-harmonic maps and inhomogeneous Landau–Lifshitz equations. Comment. Math. Helv.. 2006, 81 2 433-448

[12] Lieberman GM. Second Order Parabolic Differential Equations. 1996 River Edge: World Scientific Publishing Co., Inc

[13] Ma L. Harmonic map heat flow with free boundary. Comment. Math. Helv.. 1991, 66 2 279-301

[14] Meyer Y, Rivière T. A partial regularity result for a class of stationary Yang–Mills fields in high dimension. Rev. Mat. Iberoam.. 2003, 19 1 195-219

[15] Michor, P.W.: Topics in Differential Geometry. Graduate Studies in Mathematics, vol. 93. American Mathematical Society, Providence (2008). http://goo.gl/sbCBtN. Accessed 15 May 2017

[16] Müller F, Schikorra A. Boundary regularity via Uhlenbeck–Rivi‘ere decomposition. Analysis (Munich). 2009, 29 2 199-220

[17] Nirenberg L. An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa (3). 1966, 20 733-737

[18] Qing J. On singularities of the heat flow for harmonic maps from surfaces into spheres. Commun. Anal. Geom.. 1995, 3 1–2 297-315

[19] Qing J. A remark on the finite time singularity of the heat flow for harmonic maps. Calc. Var. Partial Differ. Equ.. 2003, 17 4 393-403

[20] Rivière T. Conservation laws for conformally invariant variational problems. Invent. Math.. 2007, 168 1 1-22

[21] Rivière T, Struwe M. Partial regularity for harmonic maps and related problems. Commun. Pure Appl. Math.. 2008, 61 4 451-463

[22] Sacks J, Uhlenbeck K. The existence of minimal immersions of 2-spheres. Ann. Math. (2). 1981, 113 1 1-24

[23] Schikorra A. A remark on gauge transformations and the moving frame method. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2010, 27 2 503-515

[24] Schoen R, Uhlenbeck K. A regularity theory for harmonic maps. J. Differ. Geom.. 1982, 17 2 307-335

[25] Schoen RM. Chern SS. Analytic aspects of the harmonic map problem. Seminar on Nonlinear Partial Differential Equations. 1984 New York: Springer. 321-358

[26] Struwe M. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv.. 1985, 60 4 558-581

[27] Tao T, Tian G. A singularity removal theorem for Yang–Mills fields in higher dimensions. J. Am. Math. Soc.. 2004, 17 3 557-593

[28] Topping P. Winding behaviour of finite-time singularities of the harmonic map heat flow. Math. Z.. 2004, 247 2 279-302

[29] Uhlenbeck KK. Connections with Lpbounds on curvature. Commun. Math. Phys.. 1982, 83 1 31-42

[30] Wang C. Biharmonic maps from R4 into a Riemannian manifold. Math. Z.. 2004, 247 1 65-87

[31] Wang C. Stationary biharmonic maps from Rm into a Riemannian manifold. Commun. Pure Appl. Math.. 2004, 57 4 419-444