Constant Mean Curvature Tori in the 3Sphere
Nicholas SchmittMedia
Description
We see embedded equivariant tori, that is embedded rectangular tori with 2, 3, 4, 5 and 13 lobes.
Further, tori of revolution and tzwizzled tori. 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 threesphere have been investigated, of which the equivariant tori are among the simplest examples.
At last, in the video section we have a flow to the Wente torus. This video shows a deformation through constant mean curvature tori, starting (after a blowup) at the threelobed Wente torus in $\mathbb{R}^3$, and ending at a doublywrapped homogeneous torus in $S^3$. The family is shown stereographically projected to $\mathbb{R}^3$.
The deformation ends at a spectral genus zero torus in $S^3$ 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 periodpreserving deformation of Kilian and Schmidt [1]. This video is a remake of one of Wjatscheslaw Kewlin's[2] pioneering videos of tori deformations to Wente tori with various lobe counts.
References

M. Kilian and M.U. Schmidt.
On the moduli of constant mean curvature cylinders of finite type in the 3sphere.
preprint, 2007.
arXiv:0712.0108. 
Wjatscheslaw Kewlin.
Deformation of constant mean curvature tori in a three sphere.
Dissertation, Universität Mannheim, 2008.