Nonfattening conjecture for cylindrical and spherical singularities

Let $F_t(M)$ denote the level set flow starting from a closed embedded surface $M$. Its singularities are called cylindrical or spherical when their tangent flows have the corresponding cylindrical or spherical models. Nonfattening conjecture. If all singularities of the level set flow are either cylindrical or spherical, then the level set flow does not fatten. Here, fattening means that $F_t(M)$ develops nonempty interior. The paper proves this conjecture using its mean convex neighborhood theorem together with the Hershkovits–White nondiscrepancy result.