Length spectra and degeneration of flat metrics

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 get a boundary 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.

36 pages

References in corpus (2)

• A characterization of short curves of a Teichmueller geodesic
• Teichmuller geodesics that do not have a limit in PMF

Cited by in corpus (11)

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• Horowitz-Randol pairs of curves in q-differential metrics
• From curves to currents
• Homotopical rigidity of polygonal billiards
• Affine equivalence and saddle connection graphs of half-translation surfaces
• Length spectra and strata of flat metrics
• Subset currents on surfaces
• The length spectrum of holomorphic quadratic differential metrics