In a pioneering paper published on JDG in 1993, Leon Simon established a powerful method to demonstrate, among other things, the validity of the following result: if a multiplicity one minimal k-dimensional surface (stationary varifold) is sufficiently close, in the unit ball and in a weak measure-theoretic sense, to the stationary cone given by the union of three k-dimensional half-planes meeting along a (k-1)-dimensional subspace and forming angles of 120 degrees with one another, then, in a smaller ball, the surface must be a C^{1,\alpha} deformation of the cone. In this talk, I will present the proof of a parabolic counterpart of this result, which applies to general classes of (possibly forced) weak mean curvature flows (Brakke flows). I will particularly focus on the apparent need of an assumption, which is absent in the elliptic case, and which, on the other hand, is satisfied by both Brakke flows with multi-phase grain boundaries structure and by Brakke flows that are flows of currents mod 3: these are the main classes of Brakke flows for which a satisfactory existence theory is currently available and triple junction singularities are expected. In these cases, the theorem implies uniqueness of multiplicity-one, backward-static triple junctions as tangent flows as well as a structure theorem on the singular set under suitable Gaussian density restrictions.
This is a joint work with Yoshihiro Tonegawa (Institute of Science Tokyo).