Elastic curves and Willmore Hopf-Tori

Whitham deformation The moduli space of closed wavelike elastic curves in S 2 .Figure: Closed wave-like elastica lie on a discrete dense set of curves. Theposition of the branchpoint in the first quadrant is shown. All1-parameter family of closed wave-like elastica start at a multiwrappedequator (along the real axis). The semicircle-like curves represent familieswhich end at multiwrapped equators, those to the left of the thick lineend in a singular limit. The thick curve marks the boundary betweenthese two types. The dashed curve represents the free elastica.