Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders

A general equation of mechanical equilibrium of fluid membranes subject to bending elasticity [reported in Phys. Rev. Lett. 59, 2486 (1987)] is derived in detail. The second variation of the shape energy, also obtained for arbitrary shapes, is used to analyze stability with respect to deformational modes for spherical and cylindrical vesicles. The former analysis is well known, while the latter is presented here for the first time. The theoretical results are shown to agree very well with previous numerical calculations. In addition, they provide the energies controlling the shape fluctuations and show that spontaneous curvature may transform cylinders into tapes or strings of beads. The study of the energy of infinitesimal deformations is finally extended to include the third variation. Applying the general result to the sphere, we obtain the critical value of spontaneous curvature below which oblate ellipsoids of a deformed sphere are more stable than prolate ones. It is shown to be the same regardless of whether volume or pressure is kept constant.