Periodic-cylinder vesicle with minimal energy

We give some details about the periodic cylindrical solution found by Zhang and Ou-Yang in 1996 {\em Phys. Rev.} E {\bf 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 \emph{mathematic}, we find that this periodic shape has the minimal total energy for one period when the period--amplitude ratio $\beta\simeq1.477$, 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).