Elastic curves and Willmore Hopf-Tori

◮ If h : S 3 → S 2 is the Hopf-fibration, then h −1 (γ) for a closedcurve γ : R → S 2 is called a Hopf-torus. The curve is calledits profile curve◮ A Hopf-torus is a constrained Willmore Hopf-torus iff itsprofile curve is a closed constrained elastic curve (area andlength constraint). Its geodesic curvature satisfiesκ ′2 + 1 4 κ4 + a κ 2 + b κ + c = 0 .◮ When b = 0 we speak of elastic curves (area constraint).◮ The torus is a Willmore Hopf-torus iff the profile curve is aclosed free elastic curve (no constraint). Its geodesiccurvature satisfiesκ ′2 + 1 4 κ4 + 1 2 κ2 + c = 0 .