Constrained Willmore Hopf Tori

Abstract

Generalized elastic curves on $\S^2$ are elliptic solutions of a differential equation on the curvature of the curve. These equations are solved in terms of Weierstrass elliptic functions depending on the parameters of the differential equation. It is investigated which of these parameters yield closed curves on $\S^2$ and how these curves can be parametrized. The Hopf fibration $h:\S^3\to\S^2$ lifts closed generalized elastic curves to tori in $\S^3$. These tori are constrained Willmore surfaces, i.e. extremal values of the Willmore functional under variations preserving the conformal structure. They are called constrained Willmore Hopf tori. The conformal class and the Willmore energy of such tori is calculated