Cylindrical equilibrium shapes of fluid membranes

TL;DR

This paper analytically derives all cylindrical equilibrium shapes of fluid membranes based on curvature models, providing explicit solutions and conditions for their geometric properties.

Contribution

It presents the complete set of solutions to the membrane shape equation for cylindrical shapes, including explicit formulas and geometric conditions.

Findings

Explicit solutions for cylindrical membrane shapes

Conditions for closed and self-intersecting shapes

Analytic expressions for position vectors

Abstract

Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian curvatures by the so-called membrane shape equation. In the present paper, all solutions to this equation determining cylindrical membrane shapes are found and presented, together with the expressions for the corresponding position vectors, in explicit analytic form. The necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given.