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Baum, Helga

Eigenvalue estimates for Dirac operators coupled to instantons. (English) Zbl 0823.34072

Ann. Global Anal. Geom. 12, No. 2, 193-209 (1994).
The author considers the Dirac operator on a four-dimensional Riemannian spin manifold of positive curvature with values in a vector bundle. A sharp lower bound for the first positive eigenvalue is obtained, and an analysis is given for the question when this lower bound is equal to the eigenvalue [Th. Friedrich, Math. Nachr. 102, 53-56 (1981; Zbl 0481.53039)].
Reviewer: T.Aktosun (Fargo)

Cited in 5 Documents

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35P15 Estimates of eigenvalues in context of PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Dirac operator on a four-dimensional Riemannian spin manifold of positive curvature; lower bound for the first positive eigenvalue

Citations:

Zbl 0481.53039
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Full Text: DOI

References:

[1] Atiyah, M.F.;Hitchin, N.J.;Singer, I.M.: Selfduality in four-dimensional Riemannian Geometry.Proc. R. Soc. London A 362 (1978), 425–461. · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[2] Baum, H.;Friedrich, Th.;Grunewald, R.;Kath, I.:Twistors and Killing Spinors on Riemannian Manifolds. Teubner-Texte Math. Bd. 124, Stuttgart/Leipzig 1991. · Zbl 0734.53003
[3] Bröcker, T.;Dieck, T.t.:Representation Theory of Compact Lie Groups. Springer Verlag, N.Y. 1985. · Zbl 0581.22009
[4] Friedrich, Th.: A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds.Math. Nachr. 102 (1981), 53–56. · Zbl 0481.53039 · doi:10.1002/mana.19811020106
[5] Friedrich, Th.: Self-duality of Riemannian Manifolds and Connections. In:Self-dual Riemannian Geometry and Instantons. Teubner-Texte Math. Bd. 34, Leipzig 1981, 56–104.
[6] Henze, R.:Einige Untersuchungen der Eigenwerte des zu einem Zusammenhang assoziierten Dirac-Operators. Diplomarbeit, Humboldt-Universität 1984.
[7] Kobayashi, S.;Nomizu, K.:Foundation of Differential Geometry. Vol II. Interscience Publ., 1969. · Zbl 0175.48504
[8] Lawson, H.B.;Michelsohn, M.-L.:Spin Geometry. Princeton Univ. Press, 1989. · Zbl 0688.57001
[9] Wallach, N.R.:Harmonic Analysis on Homogeneous Spaces. M. Decker Inc., N.Y. 1973. · Zbl 0265.22022
[10] Želobenko, D.P.:Compact Lie groups and their representations. Providence, RI: Amer. Math. Soc. 1973.
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