Length spectra of flat metrics coming from q-differentials

TL;DR

This paper extends the understanding of how length spectra of curves determine flat metrics on surfaces, showing that q-simple curves' lengths suffice for unique identification of metrics induced by q-differentials.

Contribution

It introduces q-simple curves and proves their length spectra uniquely determine Euclidean cone metrics from q-differentials, generalizing previous results.

Findings

q-simple curves generalize simple curves for q-differentials

Lengths of q-simple curves determine the flat metric uniquely

Extension of Duchin--Leininger--Rafi's result to q-differentials

Abstract

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves' lengths do we really need to know? It is a result of Duchin--Leininger--Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential.