Sfb 288 Differential Geometry and Quantum Physics

Discrete Surfaces
Darboux Transformations of Discrete Isothermic Surfaces

The darboux transformations for isothermic surfaces have a natural discrete analogon. Given a discrete isothermic surface, an initial point and an arbitrary real number, then there is a unique Darboux transformed surface that can be calculated by simply evolving a crossratio condition for two neighbouring points of the surface and their images.

If the starting surface is a cmc surface then there is still a three parameter family of Darboux transformed surfaces that are cmc too. The left picture shows a Bubboloid - a Darboux transform of the cylinder wich is again closed in one direction (it is cut along the yellow line).

The right image shows a darboux transform of a clifford torus. This one is closed in one direction too.

On the left side there is a minimal Darboux transformed of the Kathenoid.

The right one is not minimal.

Author: U. Hertrich-Jeromin, T. Hoffmann, U. Pinkall