Isoperimetric Ratios of Toroidal Dupin Cyclides

Abstract

The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2022, 1749–1765] and to Bostan and Yurkevich [Proc. AMS 150(5), 2022, 2131–2136] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In this work we investigate a generalization of the aforementioned results to the case of a rectangular Clifford torus. Like the square case, we find closed-form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to R3 and show that the corresponding function is strictly increasing. But unlike the square case, we show that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus. These results are obtained using a mixture of differential geometry and computer algebra. Moreover, the computer algebra technique in this work is more streamlined and more direct than in the aforementioned approaches, even for the most important case of the square Clifford torus.

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Cite

BibTeX RIS

BibTeX

@article{bostan2026isoperimetric,
  title = {{Isoperimetric Ratios of Toroidal Dupin Cyclides}},
  author = {Alin Bostan and Jingmin Chen and Thomas Yu and Sergey Yurkevich},
  journal = {Differential Geometry and its Applications},
  year = {2026},
  month = sep,
  url = {https://inria.hal.science/hal-04869169v2},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Isoperimetric Ratios of Toroidal Dupin Cyclides
AU  - Alin Bostan
AU  - Jingmin Chen
AU  - Thomas Yu
AU  - Sergey Yurkevich
PY  - 2026
DA  - 2026/09/13
JO  - Differential Geometry and its Applications
UR  - https://inria.hal.science/hal-04869169v2
AB  - The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2022, 1749–1765] and to Bostan and Yurkevich [Proc. AMS 150(5), 2022, 2131–2136] shows that the 3-D Euclidean shape of the square Clifford torus is uniquely determined by its isoperimetric ratio. This solves part of the still open uniqueness problem of the Canham model for biomembranes. In this work we investigate a generalization of the aforementioned results to the case of a rectangular Clifford torus. Like the square case, we find closed-form formulas in terms of hypergeometric functions for the isoperimetric ratio of its stereographic projection to R3 and show that the corresponding function is strictly increasing. But unlike the square case, we show that the isoperimetric ratio does not uniquely determine the Euclidean shape of a rectangular Clifford torus. These results are obtained using a mixture of differential geometry and computer algebra. Moreover, the computer algebra technique in this work is more streamlined and more direct than in the aforementioned approaches, even for the most important case of the square Clifford torus.
SN  - 0926-2245
SN  - 1872-6984
ER  -

Abstract

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