Remarks on KdV-type Flows on Star-Shaped Curves

TL;DR

This paper explores the geometric relationship between star-shaped planar curves and projective maps, revealing how certain Hamiltonian flows relate to the KdV equation and constructing explicit solutions using algebro-geometric methods.

Contribution

It establishes a connection between centro-affine geometry and projective geometry, linking Pinkall's flow to the Schwarzian KdV equation and providing explicit finite-gap solutions.

Findings

Projectivization induces a map between differential invariants.

Pinkall's flow relates to the Schwarzian KdV equation.

Explicit finite-gap solutions are constructed.

Abstract

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwarzian KdV equation as its projectivization. (For both flows, the curvature evolves by the KdV equation.) Using algebro-geometric methods and the relation of group-based moving frames to AKNS-type representations, we construct examples of closed solutions of Pinkall's flow associated with periodic finite-gap KdV potentials.