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Some energy properties of Yang-Mills connections

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Funds:  Supported by the Wu Wen-Tsun Key Laboratory of Mathematics at USTC of the Chinese Academy of Sciences.
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2016.08.002
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  • Author Bio:

    SHI Jixu, male, born in 1991, master. Research field: mathematical physics. E-mail: sjx1991@mail.ustc.edu.cn

  • Received Date: 07 December 2015
  • Accepted Date: 22 May 2016
  • Rev Recd Date: 22 May 2016
  • Publish Date: 30 August 2016
    Abstract

  • Abstract

    E is a vector bundle over a compact Riemannian manifold M=Mn, n≥4, and A is a Yang-Mills connection with Ln[]2 curvature FA on E. Through a mean value inequality of the density |FA|n[]2, an energy concentrate principle of sequences of solutions that have bounded energy is proved. Unless E is a flat bundle, the energy must be bounded from below by some positive constant.

    Abstract

    E is a vector bundle over a compact Riemannian manifold M=Mn, n≥4, and A is a Yang-Mills connection with Ln[]2 curvature FA on E. Through a mean value inequality of the density |FA|n[]2, an energy concentrate principle of sequences of solutions that have bounded energy is proved. Unless E is a flat bundle, the energy must be bounded from below by some positive constant.
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    • References(10)
  • [1]
    BOURGUIGNON J P, KARCHER H. Curvature operators: Pinching estimates and geometric examples[J]. Ann Sci cole Norm Sup(4), 1978, 11(1): 71-92.
    [2]
    BOURGUIGNON J P, LAWSON H B. Stability and isolation phenomena for Yang-Mills fields[J]. Comm Math Phys, 1981, 79(2): 189-230.
    [3]
    DONALDSON S K, KRONHEIMER P B. The Geometry of Four-Manifolds[M]. New York: Oxford University Press,1990.
    [4]
    GERHARDT C. An energy gap for Yang-Mills connections[J]. Comm Math Phys, 2010, 298(2): 515-522.
    [5]
    FEEHAN P M N. Energy gap for Yang-Mills connections, Ⅱ: Arbitrary closed Riemannian manifolds[DB/OL]. arXiv:1502.00668v2.
    [6]
    LI P, SCHOEN R. Lp and mean value properties of subharmonic functions on Riemannian manifolds[J]. Acta Mathematica, 1984, 153(1): 279-301.
    [7]
    MORREY C B. Multiple Integrals in the Calculus of Variations[M]. Berlin: Springer,1966.
    [8]
    UHLENBECK K K. Removable singularities in Yang-Mills fields[J]. Comm Math Phys, 1982, 83(1): 11-29.
    [9]
    UHLENBECK K K. The Chern classes of Sobolev connections[J]. Comm Maht Phys, 1985, 101: 445-457.
    [10]
    YANG B. Removable singularities for Yang-Mills connections in higher dimensions[J]. Pacific J Math, 2003, 209(2): 381-398.
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    [1]
    BOURGUIGNON J P, KARCHER H. Curvature operators: Pinching estimates and geometric examples[J]. Ann Sci cole Norm Sup(4), 1978, 11(1): 71-92.
    [2]
    BOURGUIGNON J P, LAWSON H B. Stability and isolation phenomena for Yang-Mills fields[J]. Comm Math Phys, 1981, 79(2): 189-230.
    [3]
    DONALDSON S K, KRONHEIMER P B. The Geometry of Four-Manifolds[M]. New York: Oxford University Press,1990.
    [4]
    GERHARDT C. An energy gap for Yang-Mills connections[J]. Comm Math Phys, 2010, 298(2): 515-522.
    [5]
    FEEHAN P M N. Energy gap for Yang-Mills connections, Ⅱ: Arbitrary closed Riemannian manifolds[DB/OL]. arXiv:1502.00668v2.
    [6]
    LI P, SCHOEN R. Lp and mean value properties of subharmonic functions on Riemannian manifolds[J]. Acta Mathematica, 1984, 153(1): 279-301.
    [7]
    MORREY C B. Multiple Integrals in the Calculus of Variations[M]. Berlin: Springer,1966.
    [8]
    UHLENBECK K K. Removable singularities in Yang-Mills fields[J]. Comm Math Phys, 1982, 83(1): 11-29.
    [9]
    UHLENBECK K K. The Chern classes of Sobolev connections[J]. Comm Maht Phys, 1985, 101: 445-457.
    [10]
    YANG B. Removable singularities for Yang-Mills connections in higher dimensions[J]. Pacific J Math, 2003, 209(2): 381-398.
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