We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in terms of the Ricci curvature, provided that $Y$ is not an $L$-space (in the sense of Floer homology). The latter is a purely topological condition, and holds in a variety of examples. Performing the analogous refinement in the case of manifolds with $b_1>0$, we obtain a gauge-theoretic proof of an inequality of Brock and Dunfield relating the Thurston and $L^2$ norms of hyperbolic three-manifolds, first proved using minimal surfaces.

中文翻译：

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Monopole Floer 同源性和三流形的光谱几何

我们改进了 Seiberg-Witten 理论中的一些经典估计，并讨论了在三流形谱几何中的应用。特别地，我们证明了在有理同调三球体 $Y$ 上，对于任何黎曼度量，同精确形式上的拉普拉斯算子的第一个特征值在 Ricci 曲率方面有明确的上界，前提是 $Y$ 不是一个 $L$-空间（在 Floer 同源的意义上）。后者是一个纯粹的拓扑条件，并适用于各种例子。在 $b_1>0$ 流形的情况下执行类似的改进，我们获得了 Brock 和 Dunfield 不等式的规范理论证明，该不等式涉及 Thurston 和双曲三流形的 $L^2$ 范数，首先使用最小表面。