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. 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 [2] with closing conditions adapted to CMC cylinders in E3 [4].

A straight twizzler being bent around a circle to form a torus.

References

  1. John Bolton, Franz Pedit and Lyndon Woodward, Minimal surfaces and the affine Toda field model, J. Reine Angew. Math. 459 (1995), 119–150.
  2. N. J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere, J. Differential Geom. 31 (1990), no. 3, 627–710.
  3. M. Kilian and M. U Schmidt, On the moduli of constant mean curvature cylinders of finite type in the 3-sphere, arXiv:0712:0108v2, 2008.
  4. N. Schmitt, Flowing CMC cylinders to tori, Preprint, 2008.