Advertisement

Annals of Global Analysis and Geometry

, Volume 48, Issue 3, pp 223–232 | Cite as

Inequalities of Simons type and gaps for Yang–Mills fields

  • Zhen-Rong ZhouEmail author
Article

Abstract

In this paper, we establish an inequality of Simons type for Yang–Mills fields, and obtain a gap property, which generalize the results obtained in Bourguignon and Lawson (Comm Math Phys 79(2):189–230, 1981).

Keywords

Yang–Mills field Gap 

Mathematics Subject Classification

MR2000: 58E20 

References

  1. 1.
    Bourguignon, J.P., Lawson, H.B.: Stability and isolation phenomena for Yang-Mills fields. Comm. Math. Phys. 79(2), 189–230 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourguignon, J.P., Lawson, H.B., Simons, J.: Stability and gap phenomena for Yang–Mills fields. Proc. Acad. Sci. USA 76, 1550–1553 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, Q., Zhou, Z.R.: On gap properties and instabilities of \(p\)-Yang–Mills fields. Can. J. Math. 59(6), 1245–1259 (2007)zbMATHCrossRefGoogle Scholar
  4. 4.
    Dodziuk, J., Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields over complete manifolds. Compositio Math. 47, 165–169 (1982)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. In: Proceedings of Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1990)Google Scholar
  6. 6.
    Gantmacher, F.R.: The Theory of Matrices, AMS Chelsea Publishing, American Mathematical Society Providence, Rhode Island (2000)Google Scholar
  7. 7.
    Gerhardt, C.: An energy gap for Yang–Mills connections. Comm. Math. Phys. 298, 515–522 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Jia, G.Y., Zhou, Z.R.: Stability of F-Yang–Mills fields on submanifolds. Arch. Math. 49, 125–139 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Jia, G.Y., Zhou, Z.R.: Gaps of F-Yang–Mills fields on submanifolds. Tsukuba J. Math. 36(1), 121–134 (2012)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields. Compositio Math. 47(2), 153–163 (1982)Google Scholar
  11. 11.
    Price, P.: A monotonicity formula for Yang–Mills fields. Manuscr. Math. 43, 131–166 (1983)zbMATHCrossRefGoogle Scholar
  12. 12.
    Shen, C.L.: The gap phenomena of Yang–Mills fields over the complete manifolds. Math. Z. 180, 69–77 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Uhlenbeck, K.: Connections with \(L^{p}\) bounds on curvature. Comm. Math. Phys. 83, 31–42 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Xin, Y.L.: Instability theorems of Yang–Mills fields. Acta Math. Sci. 3(1) 103–112 (1983)Google Scholar
  16. 16.
    Zhou, Z.R., Qun, C.: Global pinching lemmas and their applications to geometry of submanifolds, harmonic maps and Yang–Mills fields. Adv Math. (PRC) 32(3), 319–326Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsCentral China Normal UniversityWuhanPeople’s Republic of China

Personalised recommendations