Bogomol’nyi decomposition for vesicles of arbitrary genus

We apply the Bogomol’nyi technique, which is usually invoked in the study of solitons or models with topological invariants, to the case of elastic energy of vesicles. We show that spontaneous bending contribution caused by any deformation from metastable bending shapes falls in two distinct topological sets: shapes of spherical topology and shapes of non-spherical topology experience respectively a deviatoric bending contribution à la Fischer and a mean curvature bending contribution à la Helfrich. In other words, topology may be considered to describe bending phenomena. Besides, we calculate the bending energy per genus and the bending closure energy regardless of the shape of the vesicle. As an illustration we briefly consider geometrical frustration phenomena experienced by magnetically coated vesicles. PACS numbers: 02.40.-k, 87.16.Dg, 75.10.Hk, 11.27.+d Our motivation is to amplify on the observation of vesicles with arbitrary low genus (number of holes/handles) exhibiting conformal diffusion (spontaneous conformal transformation), namely existence of two conservation laws for vesicles [1–6]. The genus is a topological invariant: a quantity conserved under smooth transformations which does not depend on the static or dynamic equations of the system under consideration. In contrast, conformal diffusion provides evidence that the system Hamiltonian is invariant under conformal transformations. Whereas the Nœther theorem [7, 8] may be used to treat the latter invariance law in order to compute the corresponding conserved current and constant charge, the Bogomol’nyi technique enables one to treat successfully various models with topological invariants [9–11]. In this Letter we focus on the topological conservation law only; we defer the conformal diffusion to future articles. To obtain the Bogomol’nyi relationships we write down for vesicles of arbitrary genus a bending Hamiltonian as a covariant functional invariant under conformal transformations which depends on their shape only and which is suitable for the Bogomol’nyi decomposition. Applying the converse of the remarkable theorem of Gauss [12] enables us to construct such an Hamiltonian. Instead of describing shapes by their Monge representation (i.e., their surface equation) as customary [2, 3], we characterize shapes by their metric tensor and their shape tensor: the total integral of the shape tensor self-product is our successful candidate. Then the Bogomol’nyi technique reveals the topological nature of bending phenomena while differential Letter to the Editor 2 geometry extends forward the Bogomol’nyi relationships. In particular we show that any deformation of the non-trivial metastable shapes spontaneously leads to a deviatoric bending contribution à la Fischer [13–15] for shapes of spherical topology and to a mean curvature bending contribution (up to a conformal transformation of the ambient space) à la Helfrich [2, 3, 16] for shapes of non-spherical topology: our approach exhibits that the bending contribution expression depends on the shape topology —our main result concisely contained in formulae (20), (21) and (26). Before describing the bending energy of vesicles, we first succinctly recall the Bogomol’nyi technique [Eqs. (1)–(8) below] through the nonlinear σ-model [9,10,17]. More precisely we consider spin fields on curved surfaces S with the nonlinear σ-model as interaction [18–21]: