Isoperimetric Ratios of Toroidal Dupin Cyclides

TL;DR

This paper extends the study of isoperimetric ratios of Clifford tori, showing that unlike the square case, the ratio does not uniquely determine the shape for rectangular tori, with implications for biomembrane models.

Contribution

It generalizes previous results on Clifford tori by analyzing rectangular cases and demonstrates non-uniqueness of shape determination via isoperimetric ratios.

Findings

Closed-form formulas for isoperimetric ratios using hypergeometric functions.

The isoperimetric ratio is strictly increasing with respect to certain parameters.

Shape is not uniquely determined by the isoperimetric ratio for rectangular Clifford tori.

Abstract

The combination of recent results due to Yu and Chen [Proc. AMS 150(4), 2020, 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 the generalization of the aforementioned result 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 $\mathbb{R}^3$ 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.