Theoretical results on toroidal vesicles

Abstract

According to a conjecture due to Willmore [5], the toroidal surface corresponding to the absolute minimum of the curvature energy is either an axisymmetric torus of aspect ratio 1/√2 or its conformal transform. We study this problem in the context of a fluid vesicle under the constraints of constant area, i.e. constant number of lipid molecules, and of constant enclosed volume, i.e. no permeation through the membrane. We show, in particular, that, when the calculation is carried out to second order in variations normal to the surface, these constraints do not remove the conformal degeneracy [7,8]. We give the complete hierarchy of modes which selectively break all the symmetries of a torus. We show that this symmetry problem is clarified if these toroidal surfaces are represented as 2d-dimensional surfaces embedded in the hypersphere of R 4 . Our calculation demonstrates that a positive spontaneous curvature favors non-axisymmetric shapes, where the hole has moved off the center. Inspired by the Dupin Cyclide problem, we provide a variational Ansatz to give the threshold of this non-azimuthal instability as a function of the reduced volume. Finally, we discuss a new set of possible experiments to which our calculation can be applied