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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中文翻译：

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Seiberg-Witten 方程和双曲三流形的长度谱

摘要：我们展示了 Seiberg-Witten 方程不承认任何不可约解的双曲三流形的第一个例子。我们的方法在本质上依赖于双曲几何；它结合了第一个特征值的显式上限，在有理同调球体上的共存 $1$-形式 $\lambda_1^*$ 与一个版本的 Selberg 迹公式相关- 形成双曲三流形的体积和复长度谱。使用这些关系，我们还为几个双曲有理同调球体提供了 $\lambda _1^*$ 上的精确数值界限。

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