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Elastic curves and Willmore Hopf-Tori

Universidad College Cork

The pre-image of a closed curve on the 2-sphere under the Hopf-fibration is a 2-torus in the 3-sphere, a so called Hopf-torus. For such a torus to be critical for the Willmore energy, a second order ODE has to hold for the curvature of the curve, requiring it to be a free elastic curve. There is an algebrogeometric description of such elastica in terms of spectral curves. The Whitham deformations give a nice model of the moduli space of generalized wave-like elastica. In particular we show that all Willmore Hopftori are local maxima of the Willmore energy. ( Joint work with Nicholas Schmitt (Tuebingen))

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