Commuting Hamiltonian flows of curves in real space forms

TL;DR

This paper develops a geometric framework for commuting Hamiltonian flows of space curves in real space forms, directly describing the flows as vector fields on the manifold of curves with symplectic structure, revealing new insights into spectral curves and elastic curves.

Contribution

It introduces a direct geometric approach to Hamiltonian flows of space curves, linking spectral curves to geometric motions and extending the hierarchy of invariants with area and volume.

Findings

Spectral curve's real part relates to monodromy axis motion.

New explicit formula for the hierarchy of Hamiltonians.

Elastic curves characterized as solutions to an isoperimetric problem.

Abstract

Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schr\"odinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infinitesimal invariant, we describe the flows directly as vector fields on the manifold of space curves. This manifold carries a canonical symplectic form introduced by Marsden and Weinstein. Our flows are precisely the symplectic gradients of a natural hierarchy of invariants, beginning with length, total torsion, and elastic energy. There are a number of advantages to our geometric approach. For instance, the real part of the spectral curve is geometrically realized as the motion of the monodromy axis when varying total torsion. This insight provides a new explicit formula for the hierarchy of Hamiltonians. We also interpret the complex spectral curve in terms of curves in hyperbolic space and Darboux transforms. Furthermore, we complete the hierarchy of Hamiltonians by adding area and volume. These allow for the characterization of elastic curves as solutions to an isoperimetric problem: elastica are the critical points of length while fixing area and volume.