In this paper, we study the Neumann boundary value problem of the Yang-Mills $\alpha$-flow over a 4-dimensional compact Riemannian manifold with boundary. We establish the short-time existence of the Yang-Mills $\alpha$-flow in the framework of functional analysis and derive long-time existence and convergence results of classical solutions to the Yang-Mills $\alpha$-flow, provided that the $\alpha$-energy of initial connection is below some threshold. We also prove the validity of the boundary version of small energy estimates, removal of isolated singularities, and energy lower bound result for non-flat Yang-Mills connections. These results lead to the bubbling convergence of a sequence of Yang-Mills $\alpha$-connections, and as an application, we demonstrate the existence of non-trivial Yang-Mills connections with Neumann boundary.
Wanjun AI
,
Miaomiao ZHU
. THE YANG-MILLS $\alpha$-FLOW OVER 4-MANIFOLD WITH BOUNDARY[J]. Acta mathematica scientia, Series B, 2025
, 45(5)
: 2142
-2170
.
DOI: 10.1007/s10473-025-0518-7
[1] Atiyah M F, Bott R. The Yang-Mills equations over Riemann surfaces. Philos Trans Roy Soc London Ser A, 1983, 308(1505): 523-615
[2] Bourguignon J P, Lawson H B. Stability and isolation phenomena for Yang-Mills fields. Comm Math Phys1981, 79(2): 189-230
[3] Charalambous N, Gross L. The Yang-Mills heat semigroup on three-manifolds with boundary. Comm Math Phys, 2013, 317(3): 727-785
[4] Charalambous N, Gross L. Neumann domination for the Yang-Mills heat equation. J Math Phys, 2015, 56(7): Art 073505
[5] Donaldson S K. An application of gauge theory to four-dimensional topology. J Differential Geom, 1983, 18(2): 279-315
[6] Feehan P M. Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds. Adv Math, 2016, 296: 55-84
[7] Feehan P M. Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds. Adv Math, 2017, 312: 547-587
[8] Freed D S, Uhlenbeck K K.Instantons and Four-Manifolds. New York: Springer-Verlag, 1991
[9] Gritsch U. Morse theory for the Yang-Mills functional via equivariant homotopy theory. Trans Amer Math Soc, 2000, 352(8): 3473-3493
[10] Gross L. The Yang-Mills heat equation with finite action in three dimensions. Mem Amer Math Soc, 2022, 275(1349): v+111
[11] Hong M C, Tian G. Asymptotical behaviour of the Yang-Mills flow and singular Yang-Mills connections. Math Ann, 2004, 330(3): 441-472
[12] Hong M C, Tian G, Yin H. The Yang-Mills $\alpha$-flow in vector bundles over four manifolds and its applications. Comment Math Helv, 2015, 90(1): 75-120
[13] Hong M C, Yin H. On the Sacks-Uhlenbeck flow of Riemannian surfaces. Comm Anal Geom, 2013, 21(5): 917-955
[14] Kelleher C, Streets J. Singularity formation of the Yang-Mills flow. Ann Inst H Poincaré C Anal Non Linéaire, 2018, 35(6): 1655-1686
[15] Lieberman G M.Second Order Parabolic Differential Equations. London: World Scientific Publishing Co Pte Ltd, 1996
[16] Marini A. Dirichlet and Neumann boundary value problems for Yang-Mills connections. Comm Pure Appl Math, 1992, 45(8): 1015-1050
[17] Parker T H. Nonminimal Yang-Mills fields and dynamics. Invent Math, 1992, 107(2): 397-420
[18] Pulemotov A. The Li-Yau-Hamilton estimate and the Yang-Mills heat equation on manifolds with boundary. J Funct Anal, 2008, 255(10): 2933-2965
[19] Pulemotov A. Quasilinear parabolic equations and the Ricci flow on manifolds with boundary. J Reine Angew Math, 2013, 683: 97-118
[20] Sacks J, Uhlenbeck K. The existence of minimal immersions of $2$-spheres. Ann of Math, 1981, 113(1): 1-24
[21] Sadun L, Segert J. Non-self-dual Yang-Mills connections with quadrupole symmetry. Comm Math Phys, 1992, 145(2): 363-391
[22] Schlatter A. Global existence of the Yang-Mills flow in four dimensions. J Reine Angew Math, 1996, 479: 133-148
[23] Schlatter A. Long-time behaviour of the Yang-Mills flow in four dimensions. Ann Global Anal Geom, 1997, 15(1): 1-25
[24] Schoen R, Uhlenbeck K. A regularity theory for harmonic maps. J Differential Geometry, 1982, 17(2): 307-335
[25] Schoen R.Analytic aspects of the harmonic map problem//Chern S S. Seminar on Nonlinear Partial Differential Equations. New York: Springer, 1984
[26] Sedlacek S. A direct method for minimizing the Yang-Mills functional over $4$-manifolds. Comm Math Phys, 1982, 86(4): 515-527
[27] Sharp B, Zhu M M. Regularity at the free boundary for Dirac-harmonic maps from surfaces. Calc Var Partial Differential Equations, 2016, 55(2): Art 27
[28] Sibner L M, Sibner R J, Uhlenbeck K. Solutions to Yang-Mills equations that are not self-dual. Proc Nat Acad Sci, 1989, 86(22): 8610-8613
[29] Sire Y, Wei J C, Zheng Y Q.Infinite time bubbling for the $\mathrm{SU}(2)$ Yang-Mills heat flow on $\mathbb{R}^4$. arXiv: 2208.13875
[30] Solonnikov V A. Boundary value problems for linear parabolic systems of differential equations in general form. Proc Steklov Inst Math, 1965, 83: 3-163
[31] Struwe M. On the evolution of harmonic mappings of Riemannian surfaces. Comment Math Helv, 1985, 60(1): 558-581
[32] Struwe M. The Yang-Mills flow in four dimensions. Calc Var Partial Differential Equations, 1994, 2(2): 123-150
[33] Uhlenbeck K K. Connections with $L^{p}$ bounds on curvature. Comm Math Phys, 1982, 83(1): 31-42
[34] Uhlenbeck K K. Removable singularities in Yang-Mills fields. Comm Math Phys, 1982, 83(1): 11-29
[35] Waldron A. Long-time existence for Yang-Mills flow. Invent Math, 2019, 217(3): 1069-1147
[36] Wang H Y. The existence of nonminimal solutions to the Yang-Mills equation with group ${\rm SU}(2)$ on $S^2\times S^2$ and $S^1\times S^3$. J Differential Geom, 1991, 34(3): 701-767
[37] Wehrheim K. Uhlenbeck Compactness. Cambridge: European Mathematical Society, 2004
[38] Energy quantization and mean value inequalities for nonlinear boundary value problems. J Eur Math Soc, 2005, 7(3): 305-318
[39] Yin H. Direct minimizing method for Yang-Mills energy over $SO(3)$ bundle. Math Ann, 2024, 390(1): 557-587
