Yamabe Invariants and Spin^c Structures

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.Comment: Standard LaTeX fil

The Anti-Self-Dual Deformation Complex and a conjecture of Singer

Let $(M^4,g)$ be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. $(M^4,g)$ is said to be unobstructed if the cokernel of the linearization of the self-dual Weyl tensor is trivial. This condition can also be characterized as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed