Remarks on KdV-type Flows on Star-Shaped Curves
Abstract
We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into . 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.
Keywords
Cite
@article{arxiv.0808.3593,
title = {Remarks on KdV-type Flows on Star-Shaped Curves},
author = {Annalisa Calini and Thomas Ivey and Gloria Mari Beffa},
journal= {arXiv preprint arXiv:0808.3593},
year = {2010}
}
Comments
21 pages, 5 figures; revised for publication