Monopole Floer homology and the spectral geometry of three-manifolds

We refine some classical estimates in Seiberg–Witten theory, and discuss an application to the spectral geometry of three-manifolds. We show that for any Riemannian metric on a rational homology three-sphere Y, the first eigenvalue of the Hodge Laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that Y is not an L-space (in the sense of Floer homology). The latter is a computable purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with b₁ > 0, we provide a gauge-theoretic proof of an inequality of Brock and Dunfield relating the Thurston and L² norms of hyperbolic three-manifolds, first proved using minimal surfaces.