Derivation of Bending Energy of Thin, Mildly Curved, Laterally Isotropic Elastic Shell

Minimization of the energy of the system is a principle behind comprehension of stable membraneous structures, making the expression for the membrane energy a cen-tral problem in description of different systems. Here we present a derivation of the bending energy of a thin, mildly curved, laterally isotropic elastic shell from the basic assump-tions regarding the deformation and stress. Using the equations of the differential geom-etry, the bending energy is expressed in terms of the invariants of the curvature tensor: trace and determinant. It is shown which assumptions are necessary to arrive to the result which is widely used in particular in membrane biophysics for description of the shapes of membrane-enclosed microscopic structures.