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

Richard H. Bamler, Yi Lai

TL;DR

This work resolves White's Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions with cylindrical singularities by first classifying ancient, asymptotically cylindrical flows into three canonical families: round shrinking cylinders, ancient ovals, bowl solitons, and flying wing solitons (the latter arising with a Euclidean factor). The authors develop a novel leading mode condition and an induction over thresholds to obtain a Harnack-type control of differences between nearby ancient cylindrical flows, enabling a precise PDE–ODE analysis around cylindrical regions. This leads to a canonical neighborhood theorem and a uniform, quantitative version of the conjecture, together with a full parameterization of asymptotically cylindrical model flows via the map $\mathsf{Q}$ and a space decomposition that isolates unstable modes. The results provide a self-contained, global picture of singularity formation and resolution in MCF and supply foundational tools for understanding local regularity and canonical neighborhood structures near cylindrical singularities, with potential implications for higher-codimension and surgery analyses.

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.

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

TL;DR

This work resolves White's Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions with cylindrical singularities by first classifying ancient, asymptotically cylindrical flows into three canonical families: round shrinking cylinders, ancient ovals, bowl solitons, and flying wing solitons (the latter arising with a Euclidean factor). The authors develop a novel leading mode condition and an induction over thresholds to obtain a Harnack-type control of differences between nearby ancient cylindrical flows, enabling a precise PDE–ODE analysis around cylindrical regions. This leads to a canonical neighborhood theorem and a uniform, quantitative version of the conjecture, together with a full parameterization of asymptotically cylindrical model flows via the map and a space decomposition that isolates unstable modes. The results provide a self-contained, global picture of singularity formation and resolution in MCF and supply foundational tools for understanding local regularity and canonical neighborhood structures near cylindrical singularities, with potential implications for higher-codimension and surgery analyses.

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.
Paper Structure (31 sections, 39 theorems, 253 equations)

This paper contains 31 sections, 39 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Statement of main results I: Classification of asymptotically cylindrical flows
  4. Statement of main results II: The Mean Convex Neighborhood Conjecture
  5. Structure of the paper an overview of the proof
  6. Preliminaries
  7. The leading mode condition
  8. Overview and main result
  9. Terminology
  10. Flow pairs and the leading mode condition
  11. The linearized leading mode condition
  12. Proof of the induction step, Lemma \ref{['MCF-II-32:Lem_induction_step']}
  13. The leading mode condition in the case of bowls or cylinders
  14. Proof of the start of the induction, Lemma \ref{['MCF-II-32:Lem_start_induction']}
  15. Proof of the leading mode condition
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $\mathcal{M}$ be an asymptotically $(n,k)$-cylindrical mean curvature flow. Then $\mathcal{M}$ is smooth (possibly up to its extinction time), non-collapsed, convex and invariant under all ambient rotations that fix an axis parallel to $\mathbb{R}^k \times \mathbf 0^{n-k+1}$. Moreover, $\mathcal

Theorems & Definitions (113)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.8: vague version of Theorem \ref{['MCF-II-32:Thm_Vp_mode_precise']}
  • Theorem 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 103 more