Extremal curves for weighted elastic energy in surfaces of 3-space forms

A variational problem closely related to the bending energy of curves contained is surfaces of real 3-space forms is considered. We seek curves in a surface which are critical for the elastic energy when this is weighted by the total squared normal curvature energy, under two different sets of constraints: clamped curves and one free end curves of constant length. We start by deriving the first variation formula and the corresponding Euler–Lagrange equations and natural boundary conditions of these energies and characterize critical geodesics. We show how surfaces locally foliated by critical geodesics can be found by using the fundamental theorem of submanifolds. In order to find explicit solutions we classify complete rotation surfaces in a real space form for which every parallel is critical.