Constrained Willmore tori and elastic curves in 2-dimensional space forms

In this paper, we consider two special classes of constrained Willmore tori in the $3$-sphere. The first class is given by the rotation of closed elastic curves in the upper half-plane—viewed as the hyperbolic plane—around the $x$-axis. The second is given as the preimage of closed constrained elastic curves, i.e., elastic curves with enclosed area constraint, in the round $2$-sphere under the Hopf fibration. We show that all conformal types can be isometrically immersed into $S^3$ as constrained Willmore (Hopf) tori and explicitly parametrize all constrained elastic curves in $H^2$ and $S^2$ in terms of the Weierstrass elliptic functions. Furthermore, we determine the closing condition for the curves and compute the Willmore energy and the conformal type of the resulting tori.

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Published 13 May 2014