The Morphology of Vesicles of Higher Topological Genus: Conformal Degeneracy and Conformal Modes

— The morphology of vesicles with topological genus g = 2 is studied using the properties of Willmore surfaces together with the conformal invariance of the bending energy of fluid membranes. In the phase diagram of the bilayer couple model, a region exists where vesicle shapes are given by Willmore surfaces. Within this region, shapes in general have no mirror symmetries and the ground state is conformally degenerate. The conformal modes which correspond to this degeneracy can be described as closed curves in a three-dimensional space. The existence of conformal modes leads to unusual shape fluctuations which correspond to a diffusion process in shape space called conformal diffusion. Additional regions exist in the phase diagram where the ground state is unique and vesicle shapes have two or more mirror planes. Conformal degeneracy and conformal diffusion also exist in the area-difference-elasticity model.