Remarks on KdV-type flows on star-shaped curves |
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Authors: | Annalisa Calini Thomas Ivey |
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Affiliation: | a Department of Mathematics, College of Charleston, Charleston, SC 29424, USA b Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA |
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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 RP1. 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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Keywords: | 02 30Ik 02 40Dr 05 45Yv 02 10 -v |
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