Selection of toroidal shape of partially polymerized membranes

The theroretical prediction of the existence of Clifford toroidal vesicles by Ou-Yang [Phys. Rev. A 41, 4517 (1990)] has been confirmed by the experiment of Mutz and Bensimon [Phys. Rev. A 43, 4525 (1991)]. However, the few nonaxisymmetric toroidal vesicles observed by Fourcade, Mutz, and Bensimon [Phys. Rev. Lett. 68, 2551 (1992)] raise the question of why the formation of nonaxisymmetric toroidal vesicles is so rare in comparison with that of axisymmetric ones. In this report we are going to show that for a solution of the shape equation of membranes, the spontaneous curvature ${\mathit{c}}_0$ of nonaxisymmetric Dupin cyclide has to be zero, while axisymmetric Clifford solution has no such restriction as shown by Ou-Yang [Phys. Rev. A 41, 4517 (1990)].