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 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 holds true unconditionally, and it 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).