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Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} thatmeet in triples at equal angles along smooth edges and higher order junctionson lower dimensional faces is a natural extension of classical mean curvatureflow. We call such a flow a mean curvature flow with triple edges. We show thatif a smooth mean curvature flow with triple edges is weakly close to a staticunion of three n-dimensional unit density half-planes, then it is smoothlyclose. Extending the regularity result to a class of integral Brakke flows, weshow that this implies smooth short-time existence of the flow starting from aninitial surface cluster that has triple edges, but no higher order junctions.

A local regularity theorem for mean curvature flow with triple edges