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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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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
Published electronically: August 2, 2021


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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Bibliographic Information
  • Francesco Lin
  • Affiliation: Department of Mathematics, Columbia University, Room 509, MC 4406 2990 Broadway, New York, NY 10027
  • MR Author ID: 1169363
  • Email:
  • Michael Lipnowski
  • Affiliation: Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0B9
  • MR Author ID: 1140735
  • Email:
  • Received by editor(s): November 21, 2018
  • Received by editor(s) in revised form: February 13, 2020, November 30, 2020, and March 22, 2021
  • Published electronically: August 2, 2021
  • Additional Notes: The first author was partially supported by NSF grant DMS-1807242
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 35 (2022), 233-293
  • MSC (2020): Primary 57R58, 57K32
  • DOI:
  • MathSciNet review: 4322393