Monopole Floer homology and the spectral geometry of three-manifolds

TL;DR

This paper links spectral geometry and Floer homology in three-manifolds, providing bounds on eigenvalues and a gauge-theoretic proof of inequalities relating geometric norms, with implications for topology and geometry.

Contribution

It refines classical Seiberg-Witten estimates and applies them to derive bounds on Laplacian eigenvalues and inequalities in hyperbolic geometry, connecting topology and spectral analysis.

Findings

Bound on the first eigenvalue of the Laplacian on coexact one-forms in rational homology three-spheres.

A gauge-theoretic proof of an inequality relating Thurston and L2 norms in hyperbolic three-manifolds.

Explicit bounds depending on Ricci curvature for non-L-spaces.

Abstract

We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the 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 purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with $b_1>0$, we obtain a gauge-theoretic proof of an inequality of Brock and Dunfield relating the Thurston and $L^2$ norms of hyperbolic three-manifolds, first proved using minimal surfaces.