Marked length spectral rigidity for flat metrics

TL;DR

This paper proves that flat metrics on a closed surface are uniquely determined by the lengths of all closed curves, establishing a strong spectral rigidity result with a new proof approach.

Contribution

It introduces a novel proof demonstrating spectral rigidity for flat metrics, implying that identical length spectra mean the metrics are isometric.

Findings

Flat metrics are uniquely determined by their length spectra.

Two flat metrics with identical length spectra are isometric.

The proof suggests a stronger rigidity property for flat metrics.

Abstract

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a stronger rigidity result for flat metrics.