New Solutions to the Helfrich Variation Problem for the Shapes of Lipid Bilayer Vesicles: Beyond Delaunay's Surfaces

Surfaces of revolution with constant mean curvature, i.e., catenoids, unduloids, nodoids, circular cylinders, and spheres, are called Delaunay's surfaces. All these surfaces are found to be solutions of the Helfrich variation problem which is the determination of the equilibrium shapes of lipid bilayer vesicles. It is shown that two kinds of surfaces of revolution not with constant mean curvature are also rigorous solutions of the same Helfrich variation problem. The solutions are similar to unduloids and nodoids, and degenerate to spheres, circular cylinders, and tori in certain limiting cases.