We estimate spectral gaps for the Hodge norm on quadratic differentials. To each tangent direction at any point (X, q) in the principal stratum of quadratic differentials, we associate a Hodge norm, and control the logarithmic derivative of vectors perpendicular to the principal directions in terms of the q-areas of the components corresponding to thick–thin decompositions and the lengths of short curves in the q-metric. In the worst case scenario, one gets a spectral gap of size \(C_{g,n}\mathrm {sys}(X,q)^2\).

中文翻译：

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平面上的亚纯 $$L^2$$ L 2 函数

我们估计二次微分上霍奇范数的谱间隙。对于二次微分主层中任意点 ( X ,  q ) 的每个切线方向，我们关联一个 Hodge 范数，并根据对应的分量的q面积控制垂直于主方向的向量的对数导数q度量中的粗细分解和短曲线的长度。在最坏的情况下，人们会得到大小为\(C_{g,n}\mathrm {sys}(X,q)^2\)的光谱间隙。