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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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A singularity removal theorem for Yang-Mills fields in higher dimensions
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by Terence Tao and Gang Tian PDF
J. Amer. Math. Soc. 17 (2004), 557-593 Request permission

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.
References
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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
  • Email: tao@math.ucla.edu
  • Gang Tian
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 220655
  • Email: tian@math.mit.edu
  • 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.
  • © Copyright 2004 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 17 (2004), 557-593
  • MSC (2000): Primary 53C07
  • DOI: https://doi.org/10.1090/S0894-0347-04-00457-6
  • MathSciNet review: 2053951