×

zbMATH — the first resource for mathematics

A framework for Morse theory for the Yang-Mills functional. (English) Zbl 0665.58006
It is well known that if a functional defined on a non-compact Banach manifold satisfies the Palais-Smale condition then the hessian data from the set of critical points of the functional can be used to compute topological properties of the Banach manifold in question. This condition is not satisfied by the Yang-Mills functional. Even in the simplest cases this functional does not satisfy the usual hypothesis of Morse theory. However, in this paper, the author partially recovers the Morse theory for the Yang-Mills functional by studying its restriction to a countable set of finite dimensional, non-compact varieties.
The author proves that below to fixed energy E only a finite number of these varieties need be considered. In the paper it is also shown that these varieties are naturally parametrized with topological data from the 4-dimensional manifold and with the critical points of the Yang-Mills functional and that the connections on these varieties obey a priori estimates given by solutions of the Yang-Mills equations.
Reviewer: M.Tucsnak

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [A-B] Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A308, 523-615 (1982) · Zbl 0509.14014
[2] [A-J] Atiyah, M.F., Jones, J.D.S.: Topological aspects of Yang-Mills theory. Commun. Math. Phys.61, 97-118 (1978) · Zbl 0387.55009
[3] [Au] Aubin, T.: Equations différentielles non linéaire et problème de Yamabe concernant la courbure scailaire. J. Math. Pures Appl.55, 269-296 (1976) · Zbl 0336.53033
[4] [B1] Bahri, A.: The Palais-Smale condition in the Yamabe problem on an open set in ? n (Preprint)
[5] [B2] Bahri, A.: Topological methods for a certain class of functionals and application. J. Funct. Anal.41, 397-427 (1981) · Zbl 0499.35050
[6] [B-B] Bahri, A., Berestyski, H.: Forced vibrations in super quadratic Hamiltonian systems. Acta Math.152, 143-197 (1984) · Zbl 0592.70027
[7] [B-C1] Bahri, A., Coron, J.C.: Une théorie des points critiques à l’infini pour l’équation de Yamabe et le problème de Kazdan-Warner. C.R. Acad. Sci. Paris3001, 513-516 (1985) · Zbl 0585.58005
[8] [B-C2] Bahri, A., Coron, J.C.: Vers une théorie des points critiques à l’infini. Séminaire Equations aux Dérivées Partielles. Ecole Polytechnique Exposé n0 8, Nov, 1984
[9] [B-L] Bourguignon, J.P., Lawson, H.B.: Yang-Mills theory; its physical origin and differential geometric aspects. In: Yau, S.-T. (ed.) Seminar on Differential Geometry, Ann. Math. Stud.102, Princeton, 1982 · Zbl 0482.58007
[10] [B-N] Brezis, H., Nirenberg, L.: Positive solutions of non-linear elliptic equations involving critical exponents. Commun. Pure Appl. Math.36, 437-477 (1983) · Zbl 0541.35029
[11] [D1] Donaldson, S.K.: An application of gauge theory to the topology of 4-manifolds. J. Differ. Geom.18, 279-315 (1983)
[12] [D2] Donaldson, S.K.: Connections, cohomology and the intersection forms of 4-manifolds. J. Differ. Geom.24, 275-341 (1986) · Zbl 0635.57007
[13] [D3] Donaldson, S.K.: The orientation of Yang-Mills moduli spaces and 4-manifold topology. J. Differ. Geom.26, 397-428 (1987) · Zbl 0683.57005
[14] [D4] Donaldson, S.K.: Irrationality and theh-cobordism conjecture. J. Differ. Geom.26, 141-168 (1987) · Zbl 0631.57010
[15] [F-S] Fintushel, R., Stern, R.: SO(3) connections and the geometry of 4-manifolds. J. Differ. Geom.20, 523-539 (1984) · Zbl 0562.53023
[16] [F-U] Freed, U., Uhlenbeck, K.K.: Instantons and Four-Manifolds. Berlin Heidelberg New York: Springer, 1964 · Zbl 0559.57001
[17] [Ka] Kato, T.: Perturbation Theory for Linear Operators. 2nd Ed., Berlin Heidelberg New York: Springer, 1980
[18] [M-P] Marino, A., Prodi, G.: Metodi Perturbattivi nella teoria di Morse. Boll. Un. Mat. Ital.11, 1-32 (1975) · Zbl 0311.58006
[19] [Mi] Milnor, J.: Morse Theory. Princeton: Princeton University Press, 1963
[20] [Mor] Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Berlin Heidelberg New York: Springer, 1966 · Zbl 0142.38701
[21] [Pal1] Palais, R.: Critical point theory and min-max principle. Proc. Symp. Pure. Math.,15, Am. Math. Soc., Providence, RI, 1970
[22] [Pal2] Palais, R.: Ljusternik-Snirelman theory on Banach manifolds. Topology5, 115-132 (1966) · Zbl 0143.35203
[23] [Par] Parker, T.: Gauge theories on 4-dimensional Riemannian manifolds. Commun. Math. Phys.85, 563-602 (1982) · Zbl 0502.53022
[24] [S-S] Schrader, R., Seiler, R.: A uniform lower bound on the renormalized scalar Euclidean functional determinant. Commun. Math. Phys.61, 169-175 (1978) · Zbl 0449.47041
[25] [S-U] Sacks, J., Uhlenbeck, K.K.: The existence of minimal 2-spheres. Ann. Math. (2)113, 1-24 (1981) · Zbl 0462.58014
[26] [Sch] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom.20, 479-495 (1984) · Zbl 0576.53028
[27] [Sed] Sedlacek, S.: A direct method for minimizing the Yang-Mills functional over 4-manifolds. Commun. Math. Phys.86, 515-527 (1982) · Zbl 0506.53016
[28] [S-Y] Siu, Y.T., Yau, S.T.: Compact Kahler manifolds of positive bisectional curvature. Invent. Math.59, 189-204 (1980). · Zbl 0442.53056
[29] [Ta1] Taubes, C.H.: Path-connected Yang-Mills moduli spaces. J. Differ. Geom.19, 337-392 (1984) · Zbl 0551.53040
[30] [Ta2] Taubes, C.H.: Stability in Yang-Mills theories. Commun. Math. Phys.91, 235-263 (1983) · Zbl 0524.58020
[31] [Ta3] Taubes, C.H.: Monopoles and maps fromS 2 toS 2; the topology of the configuration space. Commun. Math. Phys.95, 345-391 (1984) · Zbl 0594.58053
[32] [Ta4] Taubes, C.H.: Min-max theory for the Yang-Mills-Higgs equations. Commun. Math. Phys.97, 473-540 (1985) · Zbl 0585.58016
[33] [Ta5] Taubes, C.H.: Self-dual connections on 4-manifolds with indefinite intersection matrix. J. Differ. Geom.19, 517-560 (1984) · Zbl 0552.53011
[34] [Ta6] Taubes, C.H.: Long range forces and the topology of instanton moduli spaces, Colloque in L’honneur de Laurent Schwartz, Vol. II, Astérisque,132, 243-255 (1985)
[35] [Ta7] Taubes, C.H.: Stable topology of self-dual moduli spaces. J. Differ. Geom. (to appear)
[36] [U1] Uhlenbeck, K.K.: Connections withL p-bounds on curvatures. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019
[37] [U2] Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-29 (1982) · Zbl 0491.58032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.