The epsilon-regularity theorem for Brakke flows near triple junctions

Abstract

We establish the $\varepsilon$-regularity theorem for $k$-dimensional, possibly forced, Brakke flows near a static, multiplicity-one triple junction. This result provides the parabolic analogue to L. Simon's foundational work on the singular set of stationary varifolds and confirms that the regular structure of triple junctions persists under weak mean curvature flow. The regularity holds provided the flow satisfies a mild structural assumption on its 1-dimensional slices taken orthogonal to the junction's $(k-1)$-dimensional spine, which prohibits certain topological degeneracies. We prove that this assumption is automatically satisfied by two fundamental classes of flows where such singularities are expected: codimension-one multi-phase flows, such as the canonical $\mathrm{BV}$-Brakke flows constructed by the authors, and flows of arbitrary codimension with the structure of a mod 3 integral current, which arise from Ilmanen's elliptic regularization. For such flows, therefore, the Simon type regularity holds unconditionally.