Ancient asymptotically cylindrical flows and applications

Abstract

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.

Translating surfaces under flows by sub-affine-critical powers of Gauss curvature

“…An emphasis should be given to Section 2.3 which accounts for a perspective that a solution is a one-parameter family of functions parametrized by a height variable s = log l. Then the translator can be seen as a solution to a first-order ODE in (2.31) and (2.32). Indeed, this formulation allows us to apply the Merle-Zaag's ODE Lemma A.1 in [49] which plays the crucial role in recent researches of the classification of ancient flows; [9,10,11,12,13,14,15,33,17,26,34,35,37].…”

Uniqueness and stability of singular Ricci flows in higher dimensions

“…Finally, motivated by our recent proof of the mean-convex neighborhood conjecture for mean curvature flow through neck-singularities [CHH18,CHHW19] it seems reasonable to conjecture that the canonical neighborhood property actually follows from infinitesimal properties (as opposed to global curvature assumptions such as PIC), specifically that any complete Ricci flow spacetime with compact initial condition for whose metric completion all tangent flows at singular points, c.f. [Bam20a,Bam20b], are round shrinking spherical space forms or round shrinking necks, i.e.…”

A note on blowup limits in 3d Ricci flow

“…Finally, motivated by the recent proof of the mean-convex neighborhood conjecture [11,12] and the higher-dimensional uniqueness result from Brendle-Daskalopoulos-Naff-Sesum [7], it seems likely that there is a version of Theorem 1.1 for Ricci flow through neck-singularities in higher dimensions. However, let us also remark that while blowup limits in 3d Ricci flow are always modelled on κ-solutions, quotient necks in higher dimensions can lead to new phenomena.…”