Length spectra and degeneration of flat metrics

TL;DR

This paper characterizes when a set of simple closed curves uniquely determines flat metrics on surfaces and introduces a boundary concept for these metrics via geodesic currents, revealing their degeneration to mixed structures.

Contribution

It provides a complete description of spectral rigidity for flat metrics and constructs a boundary for the space of flat metrics using geodesic currents.

Findings

Spectral rigidity depends on specific sets of curves.

A boundary for flat metrics space is constructed.

Flat metrics degenerate to mixed structures.

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.