Constrained Willmore Tori and Elastic Curves in 2-Dimensional Space Forms

TL;DR

This paper classifies and constructs special constrained Willmore tori in the 3-sphere using elastic curves in hyperbolic and spherical geometries, providing explicit formulas and energy computations.

Contribution

It introduces a new class of constrained Willmore tori derived from elastic curves, with explicit descriptions and energy calculations, expanding understanding of conformal immersions.

Findings

All conformal types can be realized as constrained Willmore tori in S^3.

Explicit formulas for elastic curves in H^2 and S^2 using elliptic functions.

Determined closing conditions and computed Willmore energies for these tori.

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 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.