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 curve 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 write down all constrained
elastic curves in H^2 and S^2 in terms of the Weierstrass elliptic functions.
Further, we determine the closing condition for the curves and compute the
Willmore energy and the conformal type of the resulting tori.Comment: 23 pages, 2 figure
