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Transactions of the American Mathematical Society

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

Author: Ursula Gritsch
Journal: Trans. Amer. Math. Soc. 352 (2000), 3473-3493
MSC (2000): Primary 58E15, 55P91
Published electronically: April 17, 2000
MathSciNet review: 1695023
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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.

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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

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
Published electronically: 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
Article copyright: © Copyright 2000 American Mathematical Society