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Morse theory for the Yang-Mills functional via equivariant homotopy theory

Author(s): Ursula Gritsch
Journal: Trans. Amer. Math. Soc. 352 (2000), 3473-3493.
MSC (2000): Primary 58E15, 55P91
Posted: April 17, 2000
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Abstract: In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds $\{ M_{2g} : g=0,1,2, \ldots \}$ with generic $SU(2)$-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group $SU(2)$ with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337-356 and Råde in Compactness theorems for invariant connections, submitted for publication.

References:

[AB]
Atiyah, M.F., Bott, R.: The Yang-Mills equation over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308 (1982), 523-615. MR 85k:14006

[AHS]
Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London A 362 (1978), 425-461. MR 80d:53023

[AS]
Atiyah, M.F., Singer, I.M.: The index of elliptic operators: III, Ann. of Math. 87 (1968), 546-604. MR 38:5245

[BoH]
Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958), 458-538. MR 21:1586

[Ba]
Braam, P.J.: Magnetic monopoles on three-manifolds, J. Differential Geometry 30 (1989), 425-464. MR 90e:53040

[Cho]
Cho, Y. S.: Finite group actions on the moduli space of self dual connections I, Trans. Amer. Math. Soc., 323, No. 1 (1991), 233-261. MR 91d:58030

[DK]
Donaldson, S.K., Kronheimer, P.B.: ``The Geometry of Four-Manifolds'', Oxford mathematical monographs, 1990, Oxford University Press. MR 92a:57036

[FU]
Freed, D.S., Uhlenbeck, K.K.: ``Instantons and Four-Manifolds'', MSRI publications, Vol. 1, 1984, New York-Berlin-Heidelberg: Springer Verlag. MR 91i:57019

[LM]
Lawson, H.B. Jr., Michelsohn, M.-L.: ``Spin Geometry'', Princeton Mathematical Series Vol. 38, 1989, Princeton University Press. MR 91g:53001

[MS]
Milnor, J.W., Stasheff, J.D.: ``Characteristic Classes'', Annals of Mathematical Studies No. 76, 1974, Princeton University Press. MR 55:13428

[Pal1]
Palais, R.S.: Lusternik-Schnirelman Theory on Banach Manifolds, Topology Vol. 5 (1966), 115-132. MR 41:4584

[Pal2]
Palais, R.S.: The Principle of Symmetric Criticality, Comm. Math. Phys. 69 (1979), 19-30. MR 81c:58026

[Pa1]
Parker, T.H.: A Morse theory for equivariant Yang-Mills, Duke Mathematical Journal, Vol. 66, No. 2 (1992), 337-356. MR 93e:58031

[Pa2]
Parker, T.H.: Non-minimal Yang-Mills fields and dynamics, Invent. Math. 107 (1992), 397-420. MR 93b:58037

[Rå]
Råde, J.: Compactness Theorems for Invariant Connections, submitted for publication.

[Se]
Segal, G.B.: Equivariant K-theory, Publ. Math. Inst. Hautes Études Sci. 34 (1968), 129-151. MR 38:2769

[SS]
Sadun, L., Segert, J.: Non-self-dual Yang-Mills connections with quadrupole symmetry, Comm. Math. Phys. 145 (1992), 363-391. MR 94b:53056

[SSU]
Sibner, L.M., Sibner, R.J., Uhlenbeck, K.K. : Solutions to Yang-Mills equations that are not self-dual, Proc. Natl. Acad. Sci. USA Vol. 86 (1989), 8610-8613. MR 90j:58032

[Wa]
Wasserman, A.G.: Equivariant differential topology, Topology Vol. 8 (1969), 127-150. MR 40:3563

[Wan]
Wang, H.-Y.: The existence of nonminimal solutions to the Yang-Mills equation with group $SU(2)$ on $S^{2} \times S^{2}$ and $S^{1} \times S^{3}$, J. Differential Geometry Vol. 34 (1991), 701-767. MR 93b:58036

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Additional Information:

Ursula Gritsch
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, U.K.
Address at time of publication: Department of Mathematics, University of California, Berkeley, Evans Hall, Berkeley, California 94720-3840
Email: ursula@math.berkeley.edu

DOI: 10.1090/S0002-9947-00-02562-9
PII: S 0002-9947(00)02562-9
Keywords: Non-minimal critical points, Yang-Mills, equivariant gauge theory, equivariant homotopy theory
Received by editor(s): March 24, 1998
Received by editor(s) in revised form: September 20, 1998
Posted: April 17, 2000
Additional Notes: This note is part of my Ph.D. thesis written at Stanford University, 1997. I thank my advisor Ralph Cohen for constant support and encouragement and the Studienstifung des deutschen Volkes for a dissertation fellowship. Part of this paper was written while the author was supported by an EPSRC-fellowship at DPMMS, Cambridge, U.K
Copyright of article: Copyright 2000, American Mathematical Society

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