Scaling of the Euler characteristic, surface area, and curvatures in the phase separating or ordering systems

We present robust scaling laws for the Euler characteristic and curvatures applicable to any symmetric system undergoing phase separating or ordering kinetics. We apply it to the phase ordering in a system of the nonconserved scalar order parameter and find three scaling regimes. The appearance of the preferred nonzero curvature of an interface separating +/- domains marks the crossover to the late stage regime characterized by the Lifshitz-Cahn-Allen scaling.