Discrete Differential Geometry studies structure preserving discretizations of the smooth theory. This is exemplified by the computational discovery of discrete compact Bonnet pairs. Since the discretization is structure preserving it was possible to recover important properties of the corresponding isothermic tori, which initiated the work on constructing smooth compact Bonnet pairs.
In this project, the researchers explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. This resolves a longstanding problem on whether the metric and mean curvature function determine a unique compact surface. The construction is based on a relationship between Bonnet pairs and isothermic surfaces. In particular, the Bonnet pairs arise from isothermic surfaces with one family of planar curvature lines.
We provide the geometry data for the Figures 1, 3, 6, 7, 8, 9 and 10 of the reference.
Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas.
Compact Bonnet Pairs: isometric tori with the same curvatures.
preprint, October 2021.
Prof. Dr. Alexander I. Bobenko +
University: TU Berlin, Institut für Mathematik, MA 881
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31424655
Prof. Dr. Tim Hoffmann +
University: TU München, Department of Mathematics, 02.06.021
Address: Boltzmannstr. 3, 85748 Garching, GERMANY
Tel: +49 89 28918384