Inventiones mathematicae

, Volume 182, Issue 2, pp 231–277

Length spectra and degeneration of flat metrics

  • Moon Duchin
  • Christopher J. Leininger
  • Kasra Rafi

DOI: 10.1007/s00222-010-0262-y

Cite this article as:
Duchin, M., Leininger, C.J. & Rafi, K. Invent. math. (2010) 182: 231. doi:10.1007/s00222-010-0262-y


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.

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Moon Duchin
    • 1
  • Christopher J. Leininger
    • 2
  • Kasra Rafi
    • 3
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of MathematicsUniversity of OklahomaNormanUSA