Length spectra and degeneration of flat metrics

Moon Duchin, Christopher J. Leininger, Kasra Rafi

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)231-277
Number of pages47
JournalInventiones Mathematicae
Issue number2
StatePublished - 2010

ASJC Scopus subject areas

  • General Mathematics


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