mKdV-Related Flows for Legendrian Curves in the Pseudohermitian 3-Sphere

Abstract

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, is 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 that evolve by rigid motions in U(2). Generalizations of our results to higher-order evolutions and curves in similar geometries are also discussed.