Yamabe Invariants and Spin^c Structures

TL;DR

This paper investigates the Yamabe invariant for certain 4-manifolds, showing it is positive but less than that of the 4-sphere, using spin^c Dirac operators in a novel, elementary approach.

Contribution

It introduces a new elementary method employing spin^c Dirac operators to estimate Yamabe invariants, complementing existing Seiberg-Witten techniques.

Findings

Yamabe invariant is positive for the studied 4-manifolds.

The invariant is strictly less than that of the 4-sphere.

The method provides an elementary alternative to Seiberg-Witten-based estimates.

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.