Scaling law analysis of crystals on curved surfaces

Curved crystals are different from their flat counterparts. Experiments where colloidal particles are put onto liquid droplets have shown that, with enough particles, the resulting crystals are highly anisotropic and fractal-like, even though they form slowly enough that energy minimization should apply. Motivated by the surrounding literature, we study a continuum model of curved crystals involving minimization of a bulk, non-Euclidean elastic energy plus a surface energy term. We establish upper and lower bounds on the scaling law of the minimum energy with respect to its parameters and identify two laws distinguishing anisotropic and isotropic crystal growth. Mathematically, the key ingredient is a new "thin isoperimetric inequality" whose optimizers are not spheres. This inequality yields lower bounds on the energy of a curved crystal, which are optimal in the expected anisotropic regime. This is joint work with Ian Tobasco.