Periodic-cylinder vesicle with minimal energy

We give some details about the periodic cylindricalsolution found by Zhang and Ou-Yang in [1996 {em Phys. Rev.} E {bf53} 4206] for the general shape equation of vesicle. Three differentkinds of periodic cylindrical surfaces and a special closedcylindrical surface are obtained. Using the elliptic functionscontained in emph{mathematic}, we find that this periodic shape hasthe minimal total energy for one period when the period--amplituderatio $betasimeq1.477$, and point out that it is a discontinuousdeformation between plane and this periodic shape. Our results alsoare suitable for DNA and multi-walled carbon nanotubes (MWNTs).