Topology Seminar (Main Talk): The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

Seminar | October 31 | 4:10-5 p.m. | 3 Evans Hall

 Francesco Lin, Princeton University

 Department of Mathematics

This is joint work with Michael Lipnowski. 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 \(\lambda∗\) 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 \(\lambda∗\) for several hyperbolic rational homology spheres.