A Plateau Problem for Membranes

We use perturbation theory and numerical computation to solve a "Plateau problem" for fluid membranes with fixed area and resistance to bending. In our version of the problem, we are given two closed curves, which may not necessarily be axisymmetric or planar, and we determine the shape of the membrane that minimizes the bending energy. Since the perturbed surfaces are not necessarily functions of the axial extension, a special coordinate frame is devised to parameterize these highly deformed shapes. The permissible surfaces are found to be generalizations of minimal surfaces such as the catenoid that characterize the shapes of soap films; these surfaces can also display hallmarks of confined elastic surfaces such as buckling and wrinkling. The connection between these two systems is made precise by a one-to-one correspondence between stability eigenvalues. Our mathematical description provides insight into the forces and torques needed to stabilize cellular membranes and other elastic interfaces.