GeometrieWerkstatt Gallery
CMC Tori

Flowing CMC Cylinders to Tori

Constant mean curvature cylinders and tori in euclidean 3-space

There is a flow through constant mean curvature (CMC) cylinders in euclidean 3-space with spectral genus 2 which reaches a dense subset of CMC tori along the way [2,1]. Starting at a twizzler (equivariant CMC surface) with a straight-line axis, and opening up a double point at the Sym point, the flow bends the straight axis into a circular “soul curve” with shrinking radius, leading to a Wente torus. The flow is as in [3] with closing conditions adapted to CMC cylinders in E3 [4].

A straight twizzler being bent around a circle to form a torus.
Flow through spectral genus two tori.

References

  1. J. Bolton, F. Pedit, and L. Woodward, Minimal surfaces and the affine Toda field model, J. Reine Angew. Math. 459 (1995), 119—150 [1319519].
  2. N. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), no. 3, 627—710 [1053342].
  3. 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.
  4. N. Schmitt, Flowing CMC cylinders to tori, Preprint (2008).