Abstract
We give some details about the periodic cylindrical solution found by Zhang and Ou-Yang in [1996 Phys. Rev. E 53 4206] for the general shape equation of vesicle. Three different kinds of periodic cylindrical surfaces and a special closed cylindrical surface are obtained. Using the elliptic functions contained in mathematic, we find that this periodic shape has the minimal total energy for one period when the period–amplitude ratio β ≈ 1.477, and point out that it is a discontinuous deformation between plane and this periodic shape. Our results also are suitable for DNA and multi-walled carbon nanotubes (MWNTs).