We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calulations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequalitys saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a mean field result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.
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