Shape deformations of spherical vesicles under radial pressure

We derive the shape deformations, which occur as bifurcations from spherically symmetric vesicles, soap bubbles, and other nondissipative systems characterized by curvature elasticity (bending energy) only. A weakly nonlinear stability theory is used to obtain the lowest critical value of the control parameter (in the present case the radial pressure difference across the surface), for which nonspherical shapes can occur. The first bifurcation obtained can be of two different types, namely transcritical or subcritical, but never supercritical depending on the spontaneous curvature of the membrane, which forms the vesicle. It is shown that the well-known results of the linear stability theory are insufficient to describe the bifurcation: the actual threshold for nonspherical structures in the presence of finite-amplitude perturbations is always lower than that obtained by the linear theory; among the deformations corresponding to l=2 spherical harmonics only those with m=0 are possible; and, although these l=2, m=0 structures are the most likely ones to occur above threshold, higher harmonic structures (e.g., l=3) cannot be ruled out on the basis of a weakly nonlinear analysis.