Bogomol'nyi decomposition for vesicles of arbitrary genus

We apply the Bogomol'nyi technique, which is usually invoked in the study o f solitons or models with topological invariants, to the case of elastic en ergy of vesicles. We show that the spontaneous bending contribution caused by any deformation from metastable bending shapes falls into two distinct t opological sets: shapes of spherical topology and shapes of non-spherical t opology experience respectively a deviatoric bending contribution a la Fisc her and a mean curvature bending contribution, la Helfrich. In other words, topology may be considered to describe bending phenomena. Besides, we calc ulate the bending energy per genus and the bending closure energy regardles s of the shape of the vesicle. As an illustration we briefly consider geome trical frustration phenomena experienced by magnetically coated vesicles.