Length spectra and strata of flat metrics

TL;DR

This paper characterizes when the length spectrum of simple closed curves uniquely determines flat metrics from quadratic differentials on surfaces, extending previous results and providing conditions based on the complexity of the stratum.

Contribution

It establishes a necessary and sufficient condition for spectral rigidity in strata with enough complexity, generalizing prior work and showing local determination by finite sets.

Findings

Length spectrum determines flat metrics in strata with sufficient complexity.

Spectral rigidity is characterized by the density of the set in the projective measured foliation space.

Flat metrics are locally determined by finite sets of closed curves.

Abstract

In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Leininger-Rafi. Specifically, for any stratum with more zeroes than the genus, the length spectrum of a set of simple closed curves determines the flat metric in the stratum if and only if the set is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the length spectrum of a finite set of closed curves.