Many thin elastic bodies, such as kale leaves and torn plastic sheets, are hyperbolic in nature due to excess growth near their edges. Often, they exhibit fractal-like patterns near their boundaries. But how does minimizing the bending energy of hyperbolic surfaces lead to these frilly shapes? Venkataramani and collaborators conjectured, based in part on numerical evidence, that these shapes arise from "branch points" — a type of singularity in the second fundamental form.
In this talk I will explain the elastic model and the notion of branch points, and describe some progress towards validating this conjecture, including the result that C^2 isometric immersions are not dense in W^{2,2} isometric immersions of hyperbolic surfaces—in stark contrast to the flat and elliptic cases.
Based on joint work with Gilad Derfner and Raz Kupferman.