The Tight Dobriner Torus

Constant mean curvature tori in euclidean 3-space

A constant mean torus in euclidean 3-space with spectral genus 3. The surface was constructed by following a period-preserving flow [2] through spectral genus 3 CMC tori in the 3-sphere, starting at a flat torus. Three double points on the spectal curve were opened up to become branch points of the genus 3 spectal curve. The flow was continued until the mean curvature became infinite, limiting to a CMC surface in euclidean 3-space.

Plastic frame model of the tight Dobriner torus. The bottom part of a figure eight is visible inside the surface at the lower right.
Top view of the tight Dobriner torus. The surface can be thought of as a multiply-stacked Wente torus.
Side view of the tight Dobriner torus. The surface has a dihedral symmetry group of order 6.

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.