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

Gritsch, Ursula

doi:10.1090/S0002-9947-00-02562-9

Morse theory for the Yang-Mills functional via equivariant homotopy theory
HTML articles powered by AMS MathViewer

by Ursula Gritsch PDF

Trans. Amer. Math. Soc. 352 (2000), 3473-3493 Request permission

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

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Peter J. Braam, Magnetic monopoles on three-manifolds, J. Differential Geom. 30 (1989), no. 2, 425–464. MR 1010167
Yong Seung Cho, Finite group actions on the moduli space of self-dual connections. I, Trans. Amer. Math. Soc. 323 (1991), no. 1, 233–261. MR 1010409, DOI 10.1090/S0002-9947-1991-1010409-2
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
Daniel S. Freed and Karen K. Uhlenbeck, Instantons and four-manifolds, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, 1991. MR 1081321, DOI 10.1007/978-1-4613-9703-8
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Richard S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115–132. MR 259955, DOI 10.1016/0040-9383(66)90013-9
Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. MR 547524
Thomas H. Parker, A Morse theory for equivariant Yang-Mills, Duke Math. J. 66 (1992), no. 2, 337–356. MR 1162193, DOI 10.1215/S0012-7094-92-06610-5
Thomas H. Parker, Nonminimal Yang-Mills fields and dynamics, Invent. Math. 107 (1992), no. 2, 397–420. MR 1144429, DOI 10.1007/BF01231895
Råde, J.: Compactness Theorems for Invariant Connections, submitted for publication.
Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452
Lorenzo Sadun and Jan Segert, Non-self-dual Yang-Mills connections with quadrupole symmetry, Comm. Math. Phys. 145 (1992), no. 2, 363–391. MR 1162804
L. M. Sibner, R. J. Sibner, and K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 22, 8610–8613. MR 1023811, DOI 10.1073/pnas.86.22.8610
Arthur G. Wasserman, Equivariant differential topology, Topology 8 (1969), 127–150. MR 250324, DOI 10.1016/0040-9383(69)90005-6
Hong-Yu Wang, The existence of nonminimal solutions to the Yang-Mills equation with group $\textrm {SU}(2)$ on $S^2\times S^2$ and $S^1\times S^3$, J. Differential Geom. 34 (1991), no. 3, 701–767. MR 1139645