We refine some classical estimates in Seiberg-Witten theory, and discuss anapplication to the spectral geometry of three-manifolds. In particular, we showthat on a rational homology three-sphere $Y$, for any Riemannian metric thefirst eigenvalue of the laplacian on coexact one-forms is bounded aboveexplicitly 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 topologicalcondition, and holds in a variety of examples. Performing the analogousrefinement in the case of manifolds with $b_1>0$, we obtain a gauge-theoreticproof of an inequality of Brock and Dunfield relating the Thurston and $L^2$norms of hyperbolic three-manifolds, first proved using minimal surfaces.
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机译：我们优化Seiberg-Witten理论中的一些经典估计，并讨论了三歧管的光谱几何形状。特别是，我们展示了一个合理的同源性三个球体$ y $，对于任何riemananian度量的riemananian公制，所谓的独立的拉普拉人的特征值是在ricci曲率方面界定的，只要$ y $不是$ l $ -pace（在浮动同源的意义上）。后者是纯粹的拓扑条件，并在各种例子中保持。在以$ B_1> 0 $的歧管的情况下执行类似的型号，我们获得了Thourck和Dunfield的不等式的衡量标准，与Thurston和L ^ 2 $ Nums的双曲线三流体的规范相关，首先使用最小的表面来证明。
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