Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen--Cahn

Abstract

When mean curvature flow evolves non-uniquely, the flow is said to fatten. The work of Ilmanen shows that any weak MCF is supported inside the fattening, and work of Hershkovits--White identified canonical weak flows supported on the boundary of the fattening, known as the outermost flows. It is natural to ask, when the flow fattens, are there weak mean curvature flows supported strictly inside the fattening? Outside of some special cases (e.g. flow from cones), this question was entirely open. We show these interior flows exist, providing a general construction for non-outermost flows as limits of solutions to the parabolic $\varepsilon$-Allen--Cahn. This gives the first examples of closed, non-trivial, non-canonical, integral Brakke motions. As part of this construction, we study the $\varepsilon$-Allen--Cahn flow from low regularity initial data, and our results demonstrate the existence of integral Brakke motions from fractal sets. This includes the existence portion of Hershkovits's work on mean curvature flow from Reifenberg sets.