We consider a one-parameter family of strictly convex hypersurfaces in R n+1 moving with speed −K α ν, where ν denotes the outward-pointing unit normal vector and α ≥ 1 n+2 . For α > 1 n+2 , we show that the flow converges to a round sphere after rescaling. In the affine invariant case α = 1 n+2 , our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in 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 ε = ε(X) > 0 such that the flow is mean convex in a space-time neighborhood of size ε 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 R 3 with entropy at most 2π/e + δ 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. 1 2 KYEONGSU CHOI, ROBERT HASLHOFER, OR HERSHKOVITS 5.2. Cap size control and asymptotics 51 6. Rotational symmetry 53 6.1. Fine expansion away from the cap 53 6.2. Moving plane method 57 7. Classification of ancient low entropy flows 61 7.1. The noncompact case 61 7.2. The compact case 64 8. Applications 65 8.1. Proof of the mean convex neighborhood conjecture 65 8.2. Proof of the uniqueness conjectures for weak flows 70 References 71

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in R n+1 for all n ≥ 3: we show that if a mean curvature flow {M t } in R n+1 has an S n−1 × R singularity at (x 0 , t 0 ), then there exists an ε = ε(x 0 , t 0 ) > 0 such that M t ∩ B(x 0 , ε) is mean-convex for all t ∈ (t 0 − ε 2 , t 0 + ε 2 ). As in the case n = 2, which was resolved by the first three authors in [CHH18], 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 [HW18], 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 ≥ 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.

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are κ-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in R 3 , and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in R 3 which is strictly convex and noncollapsed.

We prove the all-time existence of non-compact, complete, strictly convex solutions to the α-Gauss curvature flow for any positive power α.