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arXiv · 0808.3593

Remarks on KdV-type Flows on Star-Shaped Curves

Annalisa Calini·Thomas Ivey·Gloria Mari Beffa
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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.

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Annalisa Calini, Thomas Ivey, Gloria Mari Beffa. 2008-08-26. Remarks on KdV-type Flows on Star-Shaped Curves. https://doi.org/10.1016/j.physd.2009.01.007

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