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 ${\lambda }_1^{\ast }$ 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 ${\lambda }_1^{\ast }$ for several hyperbolic rational homology spheres.

61 pages, 5 plots, 1 figure. The paper is now structured so that the relevant trace formula can be treated as a black box. Its proof is moved to Appendix B and has been significantly expanded. A new Appendix A with an introduction to trace formulas has also been added. This version accepted for publication