Finite-gap Solutions of the Vortex Filament Equation, I

TL;DR

This paper explores the explicit construction of quasi-periodic solutions to the vortex filament equation, linking algebraic data to geometric properties, and provides a detailed analysis of genus one solutions and their generalizations.

Contribution

It offers a complete description of genus one solutions, including special cases, and extends the connections between algebraic data and geometry to higher genus.

Findings

Complete description of genus one solutions including Euler elastica and self-intersecting filaments.

Connections between algebro-geometric data and curve geometry established.

Generalizations of these connections to higher genus solutions.

Abstract

For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such as Euler elastica, constant torsion curves, and self-intersecting filaments. We also prove generalizations of these connections to higher genus.