Necessity of the structural slicing assumption (A6)

Ascertain whether the structural assumption (A6) can be removed from the ε-regularity theorem for Brakke flows near triple junctions—i.e., prove that the conclusions of the theorem hold for flows satisfying only (A1)–(A5) and ε-closeness to a triple junction, or show that (A6) follows automatically under those hypotheses.

Background

Assumption (A6) enforces quantitative lower bounds for mass and second moments of one-dimensional slices away from the spine, precluding certain topological degeneracies that the authors cannot rule out using only the parabolic monotonicity tools available for Brakke flows. The paper proves that (A6) is automatically satisfied for two important classes: multi-phase cluster flows in codimension one and flows of currents mod 3.

The authors indicate uncertainty about whether (A6) is truly necessary in general. Removing (A6) would yield a fully unconditional parabolic ε-regularity theory mirroring Simon’s stationary result.