Abstract
A direct method is employed to minimize the Yang-Mills functional over a 4-dimensional manifold. The limiting connection is shown to be Yang-Mills, but in a possibly new bundle. We show that a topological invariant of the bundle is preserved by the minimizing process. This implies the existence of an absolute minimum of the Yang-Mills functional in a wide class of bundles.
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References
Atiyah, M., Bott, R.: On the Yang-Mills Equations over Riemann surfaces (preprint)
Dold, A., Whitney, H.: Classification of oriented sphere bundles over a 4-complex. Ann. Math.69, 667–677 (1959)
Greub, W., Petry, H.: On the lifting of structure groups, in differential geometrical methods in mathematical physics II. In: Bleuler, Petry, Reetz (eds.). Lecture Notes in Mathematics, Vol. 616. Berlin, Heidelberg, New York: Springer 1968, 217–246
Hartshorne, R.: Algebraic geometry. Berlin, Heidelberg, New York: Springer 1977
Jaffe, A., Taubes, C.: Vortices and monopoles. Boston: Birkhäuser 1980
Lemaire, L.: Applications harmoniques de surfaces Riemanniennes. J. Diff. Geo.13, 51–78 (1978)
Morrey, C.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966
Palais, R.: Foundations of global non-linear analysis. New York: Benjamin 1968
Parker, T.: Gauge theories on four dimensional Riemannian manifolds. Ph. D. thesis, Stanford University (1980)
Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math.113, 1–24 (1981)
Samelson, H.: Topology of Lie groups. Bull. Am. Math. Soc.58, 2–37 (1952)
Schoen, R., Yau, S.-T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature. Ann. Math.110, 127–142 (1979)
Taubes, C.: Self-dual Yang-Mills connections on non-self dual 4-manifolds (to appear)
t' Hooft, G.: Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus. Commun. Math. Phys.81, 267–275 (1981)
Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31–42 (1982)
Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11–30 (1982)
Uhlenbeck, K.: Variational problems for gauge fields, annals studies. Proceddings of Special Year in Diff. Geometry, Institute for Advanced Study (1980)
Warner, F.: Foundations of differentiable manifolds and Lie groups. Illinois: Scott, Foresman and Company, 1971
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Communicated by S.-T. Yau
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Sedlacek, S. A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun.Math. Phys. 86, 515–527 (1982). https://doi.org/10.1007/BF01214887
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DOI: https://doi.org/10.1007/BF01214887