Communications in Analysis and Geometry

Volume 22 (2014)

Number 2

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

Pages: 343 – 369

DOI: https://dx.doi.org/10.4310/CAG.2014.v22.n2.a6

Author

Lynn Heller (Institut für Mathematik, Universität Tübingen, Germany)

Abstract

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.

Published 13 May 2014