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, and 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 -simple curves, for any positive integer , and show that the lengths of -simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a -differential.
Cite this article
Marissa Loving, Length spectra of flat metrics coming from -differentials. Groups Geom. Dyn. 14 (2020), no. 4, pp. 1223–1240DOI 10.4171/GGD/578