Neutral Deformation Modes of Minimal Surfaces

TL;DR

This work characterizes neutral deformation modes for minimal-surface shells by decomposing the shell energy into stretching, drilling, and bending terms and expressing deformations through a conformal map between Weierstrass representations. It derives explicit closed-form expressions for $W_s$, $W_d$, and $W_b$, and establishes a hierarchy: stretching neutrality implies drilling neutrality, which in turn implies bending neutrality, with isometries of a minimal surface being globally neutral. The authors connect neutral deformations to area-preserving Möbius transformations and Bonnet transformations, and illustrate the theory with concrete examples (e.g., Bour’s, Enneper’s surfaces) including special Goursat/Möbius cases. This framework provides a geometric-energy perspective that classifies minimal surfaces relative to a reference surface via three energy contents, suggesting soft elasticity phenomena and potential extensions beyond minimal surfaces.

Abstract

Stretching, drilling, and bending are the independent deformation modes of a thin shell, each of which has an individual energy content. When the energy content of a mode vanishes, that mode is neutral. We characterize all neutral modes of deformation of minimal surfaces into minimal surfaces. A hierarchy is found among these: a stretching neutral mode (which is an isometry) is also drilling neutral, and a drilling neutral mode is also bending neutral. Thus, all isometries of a minimal surface are globally neutral and give rise to soft elasticity. More generally, all minimal surfaces can be classified relative to a reference one in terms of three energy contents, which can be given in closed form.

Figures (4)

• Figure 1: Let two minimal surfaces $\mathscr{S}$ and $\mathscr{S}^\ast$ be represented by Weierstrass functions $F$ and $F^\ast$ with simply connected open domains $\Omega$ and $\Omega^\ast$. Let $h$ be a conformal mapping such that $\Omega^\ast=h(\Omega)$. We consider the deformation $\bm{y}$ that takes $\mathscr{S}$ into the surface $\mathscr{S}^{\ast}=\bm{y}(\mathscr{S})$ and satisfies $\bm{y}\circ\bm{r} =\bm{r}^\ast\circ h$ where $\bm{r}$ and $\bm{r}^\ast$ are the mappings produced by $F$ and $F^\ast$ respectively according to \ref{['eq:Weierstrass_representation']}.
• Figure 2: Bour's surface $\mathscr{S}$ of index $m=1$ (represented by $F=1/w$) is deformed into a surface $\mathscr{S}^\ast$ represented by $F^\ast$ in \ref{['eq:drillingNeutralF']} with $\lambda=1$ and $h$ as in \ref{['eq:moebiusSpecial']} for different values of $\alpha_0$ and complex parameters $a$, $c$. Panel (a) depicts the undeformed surface $\mathscr{S}$; panels (b) and (c) depict rotations of $\mathscr{S}$, whereas the surface $\mathscr{S}^\ast$ in panels (d), (e), and (f) shows the compound effect of a rotation and a Bonnet transformation.
• Figure 3: Enneper's surface (a) and Bour's surface of index $m=3$ (b). The latter is obtained by deforming the former with no bending energy; stretching and drilling energies are given by \ref{['eq:stretching_bending_energies']}.
• Figure 4: Bour's surface of index $m=1$ depicted in Fig. \ref{['fig:soft_elasticity_a']} is deformed via the Goursat transformation in \ref{['eq:goursat_kappa']}, for different values of the parameter $\kappa$.