Abstract
In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin–Leininger–Rafi. Specifically, for any stratum with more zeroes than the genus, the \(\Sigma \)-length-spectrum of a set of simple closed curves \(\Sigma \) determines the flat metric in the stratum if and only if \(\Sigma \) is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the \(\Sigma \)-length-spectrum of a finite set of closed curves \(\Sigma \).
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Acknowledgments
The author would like to thank his advisor Chris Leininger for all the guidance in the process of this work. The author would also like to thank the anonymous referee for the helpful comments.
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Fu, SW. Length spectra and strata of flat metrics. Geom Dedicata 173, 281–298 (2014). https://doi.org/10.1007/s10711-013-9942-2
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DOI: https://doi.org/10.1007/s10711-013-9942-2