Abstract
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 \(C_{g,n}\mathrm {sys}(X,q)^2\).
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Acknowledgements
The author would like to thank Alex Eskin, Jeremy Kahn, Kasra Rafi, and Alex Wright for conversations that provided the inspiration for this project. The author was supported by a Fields Postdoctoral Fellowship and a research fellowship from National Research University Higher School of Economics.
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Frankel, I. Meromorphic \(L^2\) functions on flat surfaces. Geom. Funct. Anal. 32, 832–860 (2022). https://doi.org/10.1007/s00039-022-00611-w
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DOI: https://doi.org/10.1007/s00039-022-00611-w