Stresses in lipid membranes

The stresses in a closed lipid membrane described by the Helfrich Hamiltonian, quadratic in the extrinsic curvature, are identified using Noether's theorem. Three equations describe the conservation of the stress tensor: the normal projection is identified as the shape equation describing equilibrium configurations; the tangential projections are consistency conditions on the stresses which capture the fluid character of such membranes. The corresponding torque tensor is also identified. The use of the stress tensor as a basis for perturbation theory is discussed. The conservation laws are cast in terms of the forces and torques on closed curves. As an application, the first integral of the shape equation for axially symmetric configurations is derived by examining the forces which are balanced along the circles of constant latitude.