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A singularity removal theorem for Yang-Mills fields in higher dimensions

Authors: Terence Tao and Gang Tian
Journal: J. Amer. Math. Soc. 17 (2004), 557-593
MSC (2000): Primary 53C07
Published electronically: April 16, 2004
MathSciNet review: 2053951
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Abstract: 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.

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Additional Information

Terence Tao
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
MR Author ID: 361755
ORCID: 0000-0002-0140-7641

Gang Tian
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
MR Author ID: 220655

Keywords: Yang-Mills fields, gauge transformation, monotonicity formula, Morrey spaces
Received by editor(s): September 27, 2002
Published electronically: April 16, 2004
Additional Notes: The first author is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.
The second author is supported by an NSF grant and a Simons fund.
Article copyright: © Copyright 2004 American Mathematical Society