Acta Mathematica

Volume 228 (2022)

Number 2

Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness

Pages: 217 – 301



Kyeongsu Choi (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Robert Haslhofer (Department of Mathematics, University of Toronto, Ontario, Canada)

Or Hershkovits (Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem, Israel)


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 $\varepsilon = \varepsilon (X) \gt 0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ 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, unitregular, 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 meancurvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen’s multiplicity‑$1$ conjecture, we conclude that for embedded $2$‑spheres the meancurvature flow through singularities is well-posed.

Received 29 October 2018

Accepted 16 August 2021

Published 1 July 2022