The compensation of Gaussian curvature in developable cones is local

We use the angular deficit scheme[1] to determine numerically the distribution of Gaussian curvature in developable cones(d-cones)[2] formed by forcing a flat elastic sheet into a circular container so that the sheet buckles. This provides a new way to confirm the vanishing of mean-curvature[3] at the rim where the sheet touches the container. This angular deficit scheme also allows us to explore the potential role of the Gauss-Bonnet theorem in explaining the mean-curvature vanishing phenomenon. The theorem's global constraint on curvature resembles the global conditions observed to be relevant for vanishing mean curvature. However, our result suggests that the Gauss-Bonnet theorem does not explain the vanishing of mean-curvature. \newline [1] V. Borrelli, F. Cazals, and J.-M. Morvan, {\sl Computer Aided Geometric Design} {\bf 20}, 319 (2003). \newline [2] E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, {\sl Nature} {\bf 401}, 46 (1999). \newline [3] T. Liang and T. A. Witten, {\sl Phys. Rev. E} {\bf 73}, 046604 (2006).