Forces on membranes with in-plane order

Abstract

An ordered configuration of a vector field on an elastic curved surface induces a long-range interaction with the Gaussian curvature of the surface through the Green function of the Laplace–Beltrami operator of the surface. This in-plane orientational order configuration gives rise to a stress inducing a competition between this effective interaction and the surface tension. In this work we obtain the shape equation of the model from the surface tension term, that is a simpler way than those previously presented in the literature. We show that the stress and therefore the induced force, is a vector field tangent to the surface whose magnitude is given by the stiffness constant times the gaussian curvature of the membrane. We found that the shape of the membrane can be either a minimal surface, a developed surface or a hyperbolic one, depending on the particular regime. Explicit results are shown in the case of the catenoid and the helicoid. Finally, we find the self-adjoint stability operator of the model and obtain its eigenvalues for the nematic catenoid.