Compact Bonnet Pairs
Alexander I. Bobenko, Tim Hoffmann, Andrew O'Shea Sageman-FurnasMedia
Description
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.
References
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Alexander I. Bobenko, Tim Hoffmann, and Andrew O. Sageman-Furnas.
Compact Bonnet Pairs: isometric tori with the same curvatures.
preprint, October 2021.
arXiv:2110.06335.