Cylindrical shapes of closed lipid bilayer structures correspond to an extreme area difference between the two monolayers of the bilayer

The shapes of extreme area difference between the outer and the inner layer (ΔA) of the closed lipid bilayer structures at fixed membrane area (A) and fixed volume (V) are determined by stating and analytically solving a variational problem for axisymmetric shapes. It is shown that the spheres with at most two different radii and the cylinder are the solutions of this variational problem. The cylinder ended by a hemisphere on each end is the shape combined from these solutions and is therefore, itself the shape of the extreme ΔA at fixed V and A. The related cylindrical shapes of stearoyl–oleoyl–phosphocholine vesicles are shown.