Geometric & Functional Analysis GAFA

, Volume 8, Issue 6, pp 965–977

Yamabe Invariants and $ Spin^c $ Structures

  • M.J. Gursky
  • C. LeBrun

Abstract.

The Yamabe invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive but strictly less than that of the 4-sphere. This is done by using \( spin^c \) Dirac operators to control the lowest eigenvalue of a perturbation of the Yamabe Laplacian. These results dovetail perfectly with those derived from the perturbed Seiberg–Witten equations [Le3], but the present method is much more elementary in spirit.

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Copyright information

© Birkhäuser Verlag, Basel 1998

Authors and Affiliations

  • M.J. Gursky
    • 1
  • C. LeBrun
    • 2
  1. 1.Dept. of Math., Indiana University, Bloomington, IN 47405, USA, e-mail: gursky@indiana.eduUS
  2. 2.Dept. of Math., SUNY Stony Brook, Stony Brook, NY 11795, USA, e-mail: claude@math.sunysb.eduUS

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