The Epsilon-Regularity Theorem for Brakke Flows Near Triple Junctions

Explore the epsilon-regularity theorem for Brakke flows near triple junctions in this 46-minute mathematical lecture. Delve into the parabolic counterpart of Leon Simon's pioneering 1993 result on minimal surfaces, which demonstrated that multiplicity one minimal k-dimensional surfaces sufficiently close to stationary cones formed by three k-dimensional half-planes meeting at 120-degree angles must be C^{1,α} deformations of the cone. Examine how this fundamental principle extends to general classes of weak mean curvature flows, including possibly forced Brakke flows, and understand the critical differences between elliptic and parabolic cases. Discover the additional assumption required in the parabolic setting that is absent in the elliptic case, and learn how this assumption is satisfied by Brakke flows with multi-phase grain boundaries structure and flows of currents mod 3. Analyze the implications for uniqueness of multiplicity-one, backward-static triple junctions as tangent flows and explore the resulting structure theorem on singular sets under suitable Gaussian density restrictions. Gain insights into current existence theory for Brakke flows and understand why triple junction singularities are expected in these mathematical frameworks through this collaborative research with Yoshihiro Tonegawa from the Institute of Science Tokyo.