Constant of motion for ideal grain growth in three dimensions

Most metallic and ceramic materials are comprised of space-filling collections of crystalline grains separated by grain boundaries. While this grain structure has been studied for more than a century, there are few rigorous results regarding its global properties available in the literature. We present a rigorous result for three-dimensional grain structures that relates the integral of the Gaussian curvature over the grain boundaries to the numbers of grains and quadruple junctions. The result is numerically verified for a grain structure consisting of periodic truncated octahedra.