Unconditional parabolic counterpart to Simon’s triple-junction regularity

Determine whether a complete, unconditional parabolic counterpart to Simon’s ε-regularity theorem for multiplicity-one triple junctions holds for k-dimensional Brakke flows (k ≥ 2) under the natural L2 mean-curvature control given by assumptions (A1)–(A5), without imposing the additional structural slicing hypothesis (A6).

Background

The paper proves an ε-regularity theorem for Brakke flows near multiplicity-one triple junctions under a structural slicing assumption (A6) that guarantees quantitative lower bounds on mass and second moments for one-dimensional slices. This assumption compensates for the lack of a parabolic analogue of a key localization estimate used by Simon in the stationary setting.

For k=1, prior work establishes unconditional regularity, but for k≥2 the authors explain that the weak L2 control on mean curvature available to Brakke flows does not enforce strong topological control of slices, making an unconditional parabolic theory uncertain. The question asks whether the full parabolic analogue of Simon’s theorem can hold without (A6).