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

Lin, Francesco; Lipnowski, Michael

doi:10.1090/jams/982

The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
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by Francesco Lin and Michael Lipnowski

J. Amer. Math. Soc. 35 (2022), 233-293

DOI: https://doi.org/10.1090/jams/982

Published electronically: August 2, 2021

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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.

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