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