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

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 1 -forms λ 1 ∗ λ _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 1 -forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise numerical bounds on λ 1 ∗ λ _1^* for several hyperbolic rational homology spheres.