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Mean curvature flow through singularities

Robert Haslhofer

TL;DR

This work surveys mean curvature flow (MCF) with a focus on singularities in $\mathbb{R}^3$ and beyond, assembling a precise theory for flow through singularities and outlining a sharp program that culminates in a complete classification of noncollapsed singularities in $\mathbb{R}^4$. It bridges the PDE perspective (level-set and Brakke weak solutions) with geometric analysis (Huisken’s monotonicity, $\varepsilon$-regularity, and mean-convexity) to show that tangent flows at singularities are typically multiplicity-one self-similar shrinkers, enabling uniqueness through neck-singularities and yielding sharp bounds on the singular set. The paper highlights major advances: the mean-convex neighborhood conjecture, the uniqueness and classification of cylindrical tangent flows, and a comprehensive taxonomy of ancient noncollapsed flows in $\mathbb{R}^3$; it then extends these ideas to higher dimensions by classifying ancient noncollapsed flows in $\mathbb{R}^4$ and describing canonical neighborhoods via explicit models such as translators and 3d-ovals. Finally, it outlines key open problems and directions, including rigidity questions for shrinkers, tangent-flow uniqueness in higher dimensions, and extending the 3D theory toward a unified higher-dimensional framework with canonical neighborhoods and generic singularities.

Abstract

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in $\mathbb{R}^3$. With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in $\mathbb{R}^4$. Finally, we provide a detailed discussion of open problems and conjectures.

Mean curvature flow through singularities

TL;DR

This work surveys mean curvature flow (MCF) with a focus on singularities in and beyond, assembling a precise theory for flow through singularities and outlining a sharp program that culminates in a complete classification of noncollapsed singularities in . It bridges the PDE perspective (level-set and Brakke weak solutions) with geometric analysis (Huisken’s monotonicity, -regularity, and mean-convexity) to show that tangent flows at singularities are typically multiplicity-one self-similar shrinkers, enabling uniqueness through neck-singularities and yielding sharp bounds on the singular set. The paper highlights major advances: the mean-convex neighborhood conjecture, the uniqueness and classification of cylindrical tangent flows, and a comprehensive taxonomy of ancient noncollapsed flows in ; it then extends these ideas to higher dimensions by classifying ancient noncollapsed flows in and describing canonical neighborhoods via explicit models such as translators and 3d-ovals. Finally, it outlines key open problems and directions, including rigidity questions for shrinkers, tangent-flow uniqueness in higher dimensions, and extending the 3D theory toward a unified higher-dimensional framework with canonical neighborhoods and generic singularities.

Abstract

We first give a general introduction to the mean curvature flow, and then discuss fundamental results established over the last 10 years that yield a precise theory for the flow through singularities in . With the aim of developing a satisfying theory in higher dimensions, we then describe our recent classification of all noncollapsed singularities in . Finally, we provide a detailed discussion of open problems and conjectures.

Paper Structure

This paper contains 11 sections, 23 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic notions and fundamental tools
  3. Basic properties
  4. Preserved quantities and convexity estimate
  5. Monotonicity formula and epsilon-regularity
  6. Weak solutions
  7. Mean curvature flow through singularities in $\mathbb{R}^3$
  8. Classification of noncollapsed singularities in $\mathbb{R}^4$
  9. Further directions and open problems
  10. Open problems for mean curvature flow in $\mathbb{R}^3$
  11. Open problems and further directions in higher dimensions

Key Result

Theorem 1.1

Let $M_0\subset \mathbb{R}^{3}$ be a closed embedded surface. If $M_0$ is convex, then the mean curvature flow $\{M_t\}_{t\in[0,T)}$ starting at $M_0$ converges to a round point.

Figures (2)

  • Figure 1: Cylinder, bowl and ancient oval.
  • Figure 2: The HIMW-translators look like oval-bowls.

Theorems & Definitions (39)

  • Theorem 1.1: Huisken's convergence theorem Huisken_convex
  • Example 1.2: neckpinch singularity AltschulerAngenentGigaAngenentVelazquez
  • Proposition 2.1: mean-convexity
  • Definition 2.2: noncollapsing ShengWangandrews1HK1
  • Theorem 2.3: noncollapsing andrews1
  • Theorem 2.4: noncollapsing and convexity HK_inscribedHK1
  • Theorem 2.5: Huisken's monotonicity formula Huisken_monotonicity
  • Theorem 2.6: $\varepsilon$-regularity brakkewhite_regularityKasaiTonegawaDPGS
  • Definition 2.7: level set flow
  • Example 2.8: fattening, White_ICMIlmanenWhiteKetoverLeeZhaoCDHS
  • ...and 29 more