On the Translationally-Invariant Solutions of the Membrane Shape Equation

Abstract

The membrane shape equation derived by Helfrich and Ou-Yang describes the equilibrium shapes of biomembranes, built by bilayers of amphiphilic molecules, in terms of the mean and Gaussian curvatures of their middle-surfaces. Here, we present a new class of translationally-invariant solutions to this equation in terms of the elliptic functions which completes the solutions found earlier. In this way, all translationally-invariant solutions to the membrane shape equation are determined. Special attention is paid to those translationally-invariant solutions of the membrane shape equation which determine closed cylindrical (tube-like) surfaces (membrane shapes). Several examples of such surfaces are presented.