Skip to main content
Log in

Min-max theory for the Yang-Mills-Higgs equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In each monopole sector there exist an infinite number of finite energy solutions to the Prasad-Sommerfield limit of the SU(2) Yang-Mills-Higgs equations on ℝ3 whose energy is greater than any finite number.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. London A308, 523 (1982)

    Google Scholar 

  2. Bott, R.: Morse theoretic aspects of Yang-Mills theory. In: Recent developments in gauge theories, Cargese, France. New York: Plenum Press 1980

    Google Scholar 

  3. Palais, R.: Critical point theory and mini-max principle. Proc. Symp. Pure Math., Vol.15, Providence, RI: American Math Society 1970

    Google Scholar 

  4. Berger, M.: Nonlinearity and functional analysis. New York: Academic Press 1977

    Google Scholar 

  5. Woo, G.: Pseudo-particle configurations in two-dimensional fragments. J. Math. Phys.18, 1264 (1977)

    Google Scholar 

  6. Taubes, C.H.: The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on ℝ3, Part I. Commun. Math. Phys.86, 257 (1982); Part II. Commun. Math. Phys.86, 299 (1982)

    Google Scholar 

  7. Groisser, D.: SU(2) Yang-Mills-Higgs theory on ℝ3. Harvard University Ph.D. Thesis, 1983

  8. Taubes, C.H.: Stability in Yang-Mills theories. Commun. Math. Phys.91, 235 (1983)

    Google Scholar 

  9. Taubes, C.H.: Monopoles and maps fromS 2 toS 2; the topology of the configuration space. Commun. Math. Phys.95, 345 (1984)

    Google Scholar 

  10. Taubes, C.H.: On the Yang-Mills-Higgs equations. Bull. Am. Math. Soc.10, 295 (1984)

    Google Scholar 

  11. Jaffe, A., Taubes, C.H.: Vortices and monopoles. Boston: Birkhäuser 1980

    Google Scholar 

  12. Groisser, D.: Integrality of the monopole number in SU(2) Yang-Mills-Higgs theories on ℝ3. Commun. Math. Phys.93, 367–378 (1984)

    Google Scholar 

  13. Bogomol'nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys.24, 449 (1976)

    Google Scholar 

  14. Taubes, C.H.: The existence of multi-monopole solutions to the non-abelian Yang-Mills-Higgs equations for arbitrarily simple gauge groups. Commun. Math. Phys.80, 343 (1981)

    Google Scholar 

  15. Hitchin, N.: Monopoles and geodesics. Commun. Math. Phys.83, 579 (1982); On the construction of monopoles. Commun. Math. Phys.89, 145 (1983)

    Google Scholar 

  16. Donaldson, S.K.: Nahm's equation and the classification of monopoles. Institute for Advanced Study, Princeton, Preprint (1984)

  17. Segal, G.: Topology of spaces of rational functions. Acta Math.143, 39 (1979)

    Google Scholar 

  18. Milnor, J.: Morse theory. Princeton: Princeton University Press 1963

    Google Scholar 

  19. Uhlenbeck, K.K.: Connections withL p-bounds on curvature. Commun. Math. Phys.83, 31 (1981)

    Google Scholar 

  20. Manton, N.S.: Complex structure of monopoles. Nucl. Phys. B135, 319 (1978)

    Google Scholar 

  21. Coleman, S.: The magnetic monopole, fifty years later. Harvard University Preprint (1982)

  22. Taubes, C.H.: Path connected Yang-Mills moduli spaces. J. Diff. Geom. (to appear)

  23. Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun. Math. Phys.86, 515 (1982)

    Google Scholar 

  24. Taubes, C.H.: The structure of static Euclidean gauge fields. Harvard University Ph.D. Thesis, 1980

  25. Ladyzhenskaya, O.A.: The mathematical theory of viscous, incompressible flow. New York: Gordon & Breach 1963

    Google Scholar 

  26. Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966

    Google Scholar 

  27. Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  28. Hamilton, R.: Nash-Moser inverse function theorem. Bull. Am. Math. Soc.7, 65 (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A. Jaffe

National Science Foundation Postdoctoral Fellow in Mathematics

Rights and permissions

Reprints and permissions

About this article

Cite this article

Taubes, C.H. Min-max theory for the Yang-Mills-Higgs equations. Commun.Math. Phys. 97, 473–540 (1985). https://doi.org/10.1007/BF01221215

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01221215

Keywords

Navigation