Ancient asymptotically cylindrical flows and applications

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in ${\mathbb{R}}^{n+1}$ for all $n\ge 3$: we show that if a mean curvature flow $\{M_t\}$ in ${\mathbb{R}}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\epsilon =\epsilon (x_0,t_0)>0$ such that $M_t\cap B(x_0,\epsilon )$ is mean-convex for all $t\in (t_0-{\epsilon }^2,t_0+{\epsilon }^2)$. As in the case $n=2$, which was resolved by the first three authors in arXiv:1810.08467, the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors, we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $n\ge 3$, smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness" - where smoothness and symmetry are established in tandem.