Bending and stretching of two-dimensional fluids and solids

Soap films, lipid membranes, and elastic sheets are often analyzed with similar (idealized) models that emphasize the geometric features of these surfaces. If deformations of these surfaces are area-preserving, simple and elegant expressions may be used to describe surface and bending energies and the corresponding equations of equilibrium. However, in general, one should make a distinction between geometric energies, as measured per unit area, and elastic energies, as measured per unit mass. I will discuss some of the differences between these types of energies, and the resulting potential difficulties and inelegancies in their mathematical descriptions.