Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 254

Closed curves in ${f R}^3$: a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the Filament Equation

P.G.Grinevich, M.U.Schmidt

If a curve in ${Bbb R^3}$ is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a $2 imes2$ matrix differential operator. This operator arose in the theory of the self-focusing Nonlinear Schr"odinger Equation.

A simple spectral characterization of Bloch varieties generating periodic solutions of the Filament Equation is obtained. We show that the method of isoperiodic deformations suggested earlier by the authors for constructing periodic solutions of soliton equations can be naturally applied to the Filament Equation.

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