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Length spectra and degeneration of flat metrics

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109-1043, USA

    Moon Duchin

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, IL, 61801, USA

    Christopher J. Leininger

  3. Department of Mathematics, University of Oklahoma, Norman, OK, 73019-0315, USA

    Kasra Rafi

Authors
  1. Moon Duchin
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  2. Christopher J. Leininger
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  3. Kasra Rafi
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Correspondence to Moon Duchin.

Additional information

The first author is partially supported by NSF grant DMS-0906086. The second author is partially supported by NSF grant DMS-0905748.

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Duchin, M., Leininger, C.J. & Rafi, K. Length spectra and degeneration of flat metrics. Invent. math. 182, 231–277 (2010). https://doi.org/10.1007/s00222-010-0262-y

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  • DOI: https://doi.org/10.1007/s00222-010-0262-y

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