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

Francesco Lin Michael Lipnowski

Differential Geometry Geometric Analysis and Geometric Topology Number Theory mathscidoc:2203.10007

Journal of the Americal Mathematical Society, 35, (1), 233-293, 2022.1
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 λ_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 λ_1^* for several hyperbolic rational homology spheres.
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@inproceedings{francesco2022the,
  title={The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds},
  author={Francesco Lin, and Michael Lipnowski},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317160442746607996},
  booktitle={Journal of the Americal Mathematical Society},
  volume={35},
  number={1},
  pages={233-293},
  year={2022},
}
Francesco Lin, and Michael Lipnowski. The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds. 2022. Vol. 35. In Journal of the Americal Mathematical Society. pp.233-293. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317160442746607996.
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