We show that for a closed hyperbolic 3-manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-form Laplacian has spectral gap vanishing exponentially fast in the volume.

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双曲流形的稳定等周比和霍奇拉普拉斯算子

我们表明，对于封闭的双曲 3 流形，作用于共精确 1 形式的霍奇拉普拉斯算子的第一特征值的大小与测地线长度和稳定换向器长度相关的等周比具有比较常数，该比较常数在多项式上取决于体积和在注入半径的下限上，改进 Lipnowski 和 Stern 的估计。我们使用这个估计来表明存在封闭双曲 3-流形的序列，其注入半径低于边界且体积趋于无穷大，其中 1 形式拉普拉斯算子的光谱间隙在体积中呈指数快速消失。