Monodromy of Darboux transformations of polarised curves

TL;DR

This work studies Darboux transforms and monodromy of polarised curves in the conformal $2$-sphere through an integrable systems lens. By employing a $1$-parameter family of flat connections and polynomial conserved quantities, the authors establish the existence of resonance points for finite type polarised curves under a non-orthogonality condition. They further show that finite type curves polarised by space-form arc-length possess a linear conserved quantity, which guarantees resonance points on possible multiple covers and leads to a corollary for constrained elastic curves. The results connect the spectral theory of Darboux transforms with global properties of space-form polarisations, offering a route to construct isothermic and constrained elastic geometries via resonance phenomena.

Abstract

We show that every finite type polarised curve in the conformal $2$-sphere with a polynomial conserved quantity admits a resonance point, under a non-orthogonality assumption on the conserved quantity. Using this fact, we deduce that every finite type curve polarised by space form arc-length in the conformal $2$-sphere admits a resonance point, possibly on a multiple cover.