On the mean curvature flow of grain boundaries

Suppose that C in n+1-dimensional Euclidean space is a closed countably n-rectifiable set whose complement consists of more than one connected component. Assume that the n-dimensional Hausdorff measure of C is finite or grows at most exponentially near infinity, but no further regularity is assumed. Examples of C are planar networks in 2D, soap bubble clusters in 3D, and co-dimension one simplicial complexes in general dimensions. Very recently, we have proved a global-in-time existence of nontrivial mean curvature flow in the sense of Brakke starting from C. There exists a finite family of open sets which move continuously with respect to the Lebesgue measure, and whose boundaries coincide with the space-time support of the mean curvature flow. One may consider the open sets as grains and the flow as the mean curvature flow of grain boundaries. In this talk, mainly the background and results are discussed and if time permits, some key concepts will be discussed. This is a joint work with Lami Kim.