Yamabe Invariants and Spin^c Structures

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, but the present method is much more elementary in spirit.