Energy scaling law for the regular cone

When one removes a sector from a circular sheet of paper and glues the edges back together, the resulting shape is approximately conical. Despite the simplicity of the setup, a proof of this 'fact' starting from basic models for thin elastic sheets has not been found yet. In this talk we investigate the scaling of the elastic energy with the sheet thickness $h$ for this setting. Our main innovation is the identification of a suitable technical assumption to work with. Namely, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a diffeomorphism of some subset of the tangent space at 0 with the whole manifold. Under this assumption, we will show that the elastic energy per unit thickness scales with $h^2|\log h|$, which is the scaling of a (regularized) cone.