Ancient low entropy flows, mean convex neighborhoods, and uniqueness

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in ${\mathbb{R}}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\epsilon =\epsilon (X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\epsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in ${\mathbb{R}}^3$ with entropy at most $\sqrt{2\pi /e}+\delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed.