Regularizing rigidifying curves to understand the low-energy deformations of thin shells

It is much harder to stretch a piece of paper than bend it. We exploit this fact to simplify the elastic energy of a thin shell. We accomplish this by extending the linear isometric displacements, displacements that do not cause stretching to lowest order, to low energy Nambu-Goldstone modes. This approach fails in an interesting way in the vicinity of ``rigidifying curves,'' curves with zero normal curvature, because half of the linear isometries are divergent there. We use a renormalization group methods to show that nonlinearities in the strain regularize these divergences. We explore the relationship between these modes and folding along curves of zero normal curvature.