Embedded Equivariant Tori

Constant mean curvature tori in the 3-sphere
Embedded rectangular tori with 2, 3, 4, 5 and 13 lobes.

Rectangular Equivariant Tori

Constant mean curvature tori in the 3-sphere
Tori of revolution.

Twizzled Tori

Constant mean curvature tori in the 3-sphere
The twizzled torus is a twist on the constant mean curvature surfaces of revolution in euclidean space, classified in 1841 by C. Delaunay. Each such surface has an associate family, which extends the rotationally symmetry to a screw motion. More recently, constant mean curvature surfaces in the three-sphere have been investigated, of which the equivariant tori are among the simplest examples.

Flow to the Wente Torus

Constant mean curvature tori in the 3-sphere
This video shows a deformation through constant mean curvature tori, starting (after a blowup) at the three-lobed Wente torus in R3, and ending at a doubly-wrapped homogeneous torus in S3. The family is shown stereographically projected to R3.
The deformation ends at a spectral genus zero torus in S3 which has two double points on the unit circle in the complex plane of the spectral parameter. At this point, the deformation jumps from spectral genus two to spectral genus zero by closing four branch points to two double points.
The flow follows the period-preserving deformation of Kilian and SchmidtÂ [2]. This video is a remake of one of Wjatscheslaw Kewlin'sÂ [1] pioneering videos of tori deformations to Wente tori with various lobe counts.

References
	1.	W. Kewlin, Deformation of constant mean curvature tori in a three sphere, UniversitÃ¤t Mannheim (2008).
	2.	M. Kilian and M. Schmidt, On the moduli of constant mean curvature cylinders of finite type in the 3-sphere, arXiv:0712:0108v2 (2008) link.