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Isoperimetric Ratios of Toroidal Dupin Cyclides

Alin Bostan, Thomas Yu, Sergey Yurkevich

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.

Isoperimetric Ratios of Toroidal Dupin Cyclides

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 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.

Paper Structure

This paper contains 6 sections, 11 theorems, 46 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Properties of Toroidal Dupin cyclides and \ref{['thm:Main2']}
  3. Maxwell ratios and proofs of Theorem \ref{['thm:Thm2']} and \ref{['thm:Thm3']}
  4. Hypergeometric representations and proof of Theorem \ref{['thm:Main']}
  5. Hypergeometric representation of $A_R$ and $V_R$
  6. ${\rm Iso}_R$ is increasing

Key Result

Theorem 1.1

The 3-D Euclidean shape of $C_{\pi/4}$ is uniquely determined by its isoperimetric ratio.

Figures (3)

  • Figure 1: The circles $\mathcal{C}(\varrho) := \mathcal{C}(\varrho; R)$ for a fixed $R \in (1,\infty)$. Note that $\mathcal{C}(\varrho; R) = \mathcal{C}( (R^2-1)/\varrho ; R)$ and $\{\mathcal{C}(\varrho; R): \varrho \in [0,\sqrt{R^2-1}]\}$ partitions the half-$\rho$-$z$-plane $\rho\geqslant 0$, so $\mathcal{T}(\varrho; R) = \mathcal{T}( (R^2-1)/\varrho ; R)$, and $\{ \mathcal{T}(\varrho; R): \varrho \in [0,\sqrt{R^2-1}]\}$ partitions ${{\mathbb R}}^3$. Notice also that $\mathcal{T}(R\pm 1; R) = T_R$ = the torus of revolution generated by revolving the green circle about the $z$-axis.
  • Figure 2: Isoperimetric ratio of $i_\varrho(T_R)$. When $R \neq \sqrt{2}$, all the shapes in $\{i_\varrho(T_R): \varrho \in [0,\sqrt{R^2-1}]$ are distinct. When $R=\sqrt{2}$, only the shapes in $\{i_\varrho(T_{\sqrt{2}}): \varrho \in [0,\sqrt{2}-1]\}$ are distinct.
  • Figure 3: The shape space of 3-D torodial Dupin cyclides reduces to 2-D (circle) inversions of circle pairs (which are cross-sections of the cyclides), which further reduces to calculations involving 1-D inversions.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 2.1: Proven in YuChen:UniquenessAMS
  • Theorem 2.2: Stated in YuChen:UniquenessAMS, proven in \ref{['sec:Theorem2and3']}
  • Theorem 2.3: Stated in YuChen:UniquenessAMS, proven in \ref{['sec:Theorem2and3']}
  • Corollary 2.4
  • proof : Proof of \ref{['thm:Main2']}
  • Lemma 3.1: Equivalent to Lemma 2.3 in YuChen:UniquenessAMS
  • ...and 7 more