Energy scaling 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. This might seem very obvious, but a rigorous proof is lacking. In this talk we investigate the scaling of the elastic energy with the sheet thickness $h$ in this setting. Thinking of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space, we will 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, I will show that the elastic energy per unit thickness scales with $h^2|\log h|$, which is the scaling of a (regularized) cone.