The scalar curvature and the spectrum of the Laplacian of spin manifolds. (English) Zbl 0634.58036

The Dirac operator plays an important role in geometry of spin manifolds. For example, Lichnerowicz deduced the Â-genus vanishing theorem for spin manifolds of positive scalar curvature. In this note we give an upper bound of the greatest lower bound of the spectrum of the Laplacian on universal covering spaces of closed spin manifolds of nonvanishing Â-genus.
Reviewer: K.Ono


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
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[1] Atiyah, M.F.: Elliptic operators, discrete groups and von-Neumann algebras. Astérisque32/33, 43-72 (1976) · Zbl 0323.58015
[2] Atiyah, M.F., Singer, I.: The index of elliptic operators. III. Ann. Math.87, 546-604 (1968) · Zbl 0164.24301
[3] Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampère equations. Berlin Heidelberg New York:Springer 1982 · Zbl 0512.53044
[4] Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv.56, 581-598 (1981) · Zbl 0495.58029
[5] Berline, N., Vergne, M.: A computation of equivarint index of the Dirac operator. Bull. Soc. Math. France113, 305-345 (1985) · Zbl 0592.58044
[6] Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of non-negative Ricci curvature. J. Differ. Geom.6, 119-128 (1971) · Zbl 0223.53033
[7] Gromov, M., Lawson, H.B., Jr: Spin and scalar curvature in the presence of a fundamental group, I. Ann. Math.111, 209-230 (1980) · Zbl 0445.53025
[8] Kazdan, J., Warner, F.: Prescribing curvatures. Proc. Symp. Pure Math.27, (Part. 2), 309-319 (1975) · Zbl 0313.53017
[9] Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris Ser. A-B257, 7-9 (1963)
[10] McKean, H.P.: An upper bound to the spectrum of ? on a manifold of negative curvature. J. Differ. Geom.4, 359-366 (1970) · Zbl 0197.18003
[11] Milnor, J.: A note on curvature and the fundamental group. J. Differ. Geom.2, 1-8 (1968) · Zbl 0162.25401
[12] O’Neill, B.: The fundamental equation of a submersion. Mich. Math. J.13, 459-469 (1966) · Zbl 0145.18602
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