Geodesic
A singularity removal theorem for Yang-Mills fields in higher dimensions
Tao, Terence1 ; Tian, Gang2
1Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Journal of the American Mathematical Society, Tome 17 (2004) no. 3, pp. 557-593
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In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price’s monotonicity formula, which allows us to assert that the curvature is small not only in the $L^2$ norm, but also in the Morrey norm $M_2^{n/2}$. The second ingredient is a new inductive (averaged radial) gauge construction and truncation argument which allows us to approximate a singular gauge as a weak limit of smooth gauges with curvature small in the Morrey norm. The second ingredient is a variant of Uhlenbeck’s lemma, allowing one to place a smooth connection into the Coulomb gauge whenever the Morrey norm of the curvature is small; This variant was also proved independently by Meyer and Rivière. It follows easily from this variant that a $W^{1,2}$-connection can be placed in the Coulomb gauge if it can be approximated by smooth connections whose curvatures have small Morrey norm.
DOI : 10.1090/S0894-0347-04-00457-6

Tao, Terence  1   ; Tian, Gang  2

1 Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 
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Tao, Terence; Tian, Gang. A singularity removal theorem for Yang-Mills fields in higher dimensions. Journal of the American Mathematical Society, Tome 17 (2004) no. 3, pp. 557-593. doi: 10.1090/S0894-0347-04-00457-6

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[3] Taylor, Michael E. Partial differential equations. III 1997

[4] Tian, Gang Gauge theory and calibrated geometry. I Ann. of Math. (2) 2000 193 268

[5] Uhlenbeck, Karen K. Connections with 𝐿^{𝑝} bounds on curvature Comm. Math. Phys. 1982 31 42

[6] Uhlenbeck, Karen K. Removable singularities in Yang-Mills fields Comm. Math. Phys. 1982 11 29

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