Length spectra and degeneration of flat metrics
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- Duchin, M., Leininger, C.J. & Rafi, K. Invent. math. (2010) 182: 231. doi:10.1007/s00222-010-0262-y
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In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.