On shell membranes of Enneper type: generalized Dupin cyclides

Abstract

It is demonstrated that a class of generalized Dupin cyclides arises naturally out of a classical system of equilibrium equations for shell membranes. This class consists of all families of parallel canal surfaces on which the lines of curvature are planar. Various examples of viable membrane geometries such as particular L-minimal surfaces are presented.

Polynomial conserved quantities of Lie applicable surfaces

“…We shall now recover the result of [38,Section 6] that L-isothermic surfaces are the Combescure transforms of minimal surfaces. Let x : Σ → R 3 be an L-isothermic surface.…”

Discrete Ω$\Omega$‐nets and Guichard nets via discrete Koenigs nets

“…The same is true for $$n,\check{n}$$ when $$x$$ is $$L$$‐isothermic as in Section 7.2.1. We conclude that $$x$$ is $$L$$‐isothermic if and only if it is a Combescure transform of a minimal net $$(\check{n},n)$$ as was observed in the smooth case by Schief–Szereszewski–Rogers [42, §6].…”

“…[40, §5(b)(ii)]) An isothermic net $$x$$ with Christoffel dual $$\check{x}$$ comprise an $$O$$‐system with $$W=\mathbb {R}^{1,1}$$ and $$$$\begin{equation*} (g_{\alpha \beta })= \def\eqcellsep{&}\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \end{equation*}$$$$thanks to Theorem 4.11.The same is true for $$n,\check{n}$$ when $$x$$ is $$L$$‐isothermic as in Section 7.2.1. We conclude that $$x$$ is $$L$$‐isothermic if and only if it is a Combescure transform of a minimal net $$(\check{n},n)$$ as was observed in the smooth case by Schief–Szereszewski–Rogers [42, §6]. (c.f. [40, §5(b)(v)]) Let $$(x,n)$$ be a Guichard net with associate net $$\check{x}$$.…”

Nonlinear elastodynamics of materials with strong ellipticity condition: Carroll-type solutions

“…Remarkably, such parallel cyclide geometries arise both experimentally and in theoretical elastic membrane models of smectic A liquid crystal deformation [24][25][26][27][28][29].…”