Forces on membranes with in-plane order

Anordered configuration of a vector field on an elastic curved surface induces a long-range interaction with theGaussian curvature of the surface through theGreen 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 workwe obtain the shape equation of themodel 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 vectorfield tangent to the surface whosemagnitude is given by the stiffness constant times the gaussian curvature of themembrane.We found that the shape of themembrane can be either aminimal 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, wefind the self-adjoint stability operator of the model and obtain its eigenvalues for the nematic catenoid.