The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

TL;DR

This paper demonstrates the existence of hyperbolic three-manifolds with no irreducible solutions to the Seiberg-Witten equations, using hyperbolic geometry and spectral analysis to establish bounds and relationships.

Contribution

It provides the first examples of such manifolds and introduces a novel approach combining eigenvalue bounds with the Selberg trace formula.

Findings

Identified hyperbolic three-manifolds with no irreducible Seiberg-Witten solutions.

Established explicit bounds for the first eigenvalue on coexact 1-forms.

Linked the spectrum of the Laplacian to the volume and length spectrum of manifolds.

Abstract

We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact $1$-forms $\lambda_1^*$ on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact $1$-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise numerical bounds on $\lambda_1^*$ for several hyperbolic rational homology spheres.