Willmore tori with umbilic lines

1993 Babich and Bobenko proved in Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. Journal 72 (1993), no. 1, 151-185 the existence of Willmore tori with umbilic lines. The following example of a Willmore torus with umbilic lines is presumably the simplest one constructible by the formulas they provided. Nevertheless the shape is rather complicated. The torus itself consists of eight congruent pieces which are glued together using elementary rotations and reflections. Fig.5 shows the umbilic line and part of the torus. Since the torus intersects itself nearly parallel along this line, it makes no sense to show a picture of the whole torus. Fig.1 shows two congruent pieces already glued together. The resulting surface is topologically a cylinder and one needs four copies to get the whole torus. This is done by first rotating a second copy of the surface shown in Fig. 1 byaround the z-axis and then reflecting these two copies again at the xy-plane. Once one understands Fig.3 and Fig.4, one should have a good picture of the whole torus.

Figure 1: One quarter of the Willmore torus. One needs three more congruent copies of it to build the whole torus. This piece already has a twofold rotational symmetry around the z-axis and is topologically a cylinder. The boundary curves are displayed in Fig. 2. One of the boundary curves resembles a tennis ball curve. We call this oneand the more flat one.

Figure 2: The boundary curvesand.

Figure 3: Two copies analytically glued together alonggive half of the torus. the torus. The second copy arises from the first by a rotation ofaround the z-axis. The slightly darker strip shows a neighborhood of. Alongit twists twice. This piece has as boundary two copies of. This surface has a fourfold symmetry. To get the torus, an additional reflection at the xy-plane is needed.

Figure 4: Two copies analytically glued together alonggive half of the torus. The second copy arises from the first by a rotation ofaround the z-axis followed by a reflection at the xy-plane. The slightly darker strip shows a neighborhood of. This piece has as boundary two copies ofand only a twofold symmetry.To get the torus, an additional rotation around the z-axis is needed.