GeometrieWerkstatt Gallery
CMC Tori

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 [3] through spectral genus 3 CMC tori in the 3-sphere, starting at a flat torus [2,3,1]. 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. 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. 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.