Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

TL;DR

This paper proves the Mean Convex Neighborhood Conjecture for mean curvature flows with cylindrical singularities across all dimensions, classifies ancient asymptotically cylindrical flows, and introduces new analytical techniques.

Contribution

It provides a complete classification of ancient cylindrical flows, establishes the conjecture in full generality, and develops novel methods like the leading mode condition and induction over thresholds.

Findings

Proved the conjecture for all dimensions and cylindrical singularities.

Classified all ancient asymptotically cylindrical flows as specific canonical types.

Developed new analytical tools and a parameterization of asymptotically cylindrical flows.

Abstract

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel \emph{leading mode condition,} together with a new ``induction over thresholds'' argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.