Lipid membranes with an edge

Consider a lipid membrane with a free exposed edge. The energy describing this membrane is quadratic in the extrinsic curvature of its geometry; that describing the edge is proportional to its length. In this note we determine the boundary conditions satisfied by the equilibria of the membrane on this edge, exploiting variational principles. The derivation is free of any assumptions on the symmetry of the membrane geometry. With respect to earlier work for axially symmetric configurations, we discover the existence of an additional boundary condition which is identically satisfied in that limit. By considering the balance of the forces operating at the edge, we provide a physical interpretation for the boundary conditions. We end with a discussion of the effect of the addition of a Gaussian rigidity term for the membrane.Comment: 8 page

The one-loop elastic coefficients for the Helfrich membrane in higher dimensions

Using a covariant geometric approach we obtain the effective bending couplings for a 2-dimensional rigid membrane embedded into a $(2+D)$-dimensional Euclidean space. The Hamiltonian for the membrane has three terms: The first one is quadratic in its mean extrinsic curvature. The second one is proportional to its Gaussian curvature, and the last one is proportional to its area. The results we obtain are in agreement with those finding that thermal fluctuations soften the 2-dimensional membrane embedded into a 3-dimensional Euclidean space.Comment: 9 page