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We investigate geometric evolution equations for Legendrian curves in the 3-sphere which are invariant under the action of the unitary group U (2). We define a natural symplectic structure on the space of Legendrian loops and show that the modified Korteweg-de Vries equation, along with its associated hierarchy, are realized as curvature evolutions induced by a sequence of Hamiltonian flows. For the flow among these that induces the mKdV equation, we investigate the geometry of solutions which evolve by rigid motions in U (2). Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.
Calini et al. (Tue,) studied this question.