Vesicles of toroidal topology

We consider fluid vesicles of toroidal topology. Minimization of the curvature energy at fixed volume and area leads to three different branches of axisymmetric shapes. By using conformal transformations, we identify a large region of nonaxisymmetric shapes in the phase diagram. For vanishing spontaneous curvature, the ground state is twofold degenerate in this region and corresponds to zero pressure difference across the membrane. The relation of these results to the recent observation of toroidal shapes for partially polymerized vesicles is discussed.