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

2018/10/15 by Francesco Lin, Lin, Francesco, Michael Lipnowski +1 · 2 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.1810.06346

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 λ₁^* 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 λ₁^* for several hyperbolic rational homology spheres.

Citations

• On L-spaces and left-orderable fundamental groups
• The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Cited by

• On fixed points of pseudo-Anosov maps
• Dirac spectral flow and Floer theory of hyperbolic three-manifolds

Related

• Remarks on Sp(1)-Seiberg-Witten equation over 3-manifolds
• Homology cobordism and the geometry of hyperbolic three-manifolds
• Monopole Floer homology and the spectral geometry of three-manifolds
• Counting hyperbolic 4-manifolds with vanishing Seiberg-Witten invariants
• The Seiberg-Witten theory of homology 3-spheres