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Classification of ancient noncollapsed flows in $\mathbb{R}^4$

Kyeongsu Choi, Robert Haslhofer

TL;DR

The paper achieves a complete classification of ancient noncollapsed mean curvature flow solutions in $\mathbb{R}^4$, showing they must be among classical shrinkers, 3D bowls/ovals, or members of one-parameter translator/oval families (including HIMW and DH_ovals). A cornerstone is the differential neck theorem, which detects a small yet crucial slope term in mixed bubble-sheet convergence via a differential Merle-Zaag framework, anisotropic barriers, and propagation of smallness. The results imply that any noncompact strictly convex ancient noncollapsed flow is self-similarly translating, ruling out exotic ovals and yielding a canonical neighborhood description for mean-convex flows in $\mathbb{R}^4$. The work closes the classification program in four dimensions by separating bounded from unbounded bubble-sheet scales and delivering a precise translator-ruled asymptotic geometry with strong barrier-based control.

Abstract

In this paper, we classify all noncollapsed singularities of the mean curvature flow in $\mathbb{R}^4$. Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely $\mathbb{R}^j\times S^{3-j}$, $\mathbb{R}\times $2d-bowl, $\mathbb{R}\times $2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of $\mathbb{Z}_2\times \mathrm{O}_2$-symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of $\mathbb{Z}_2^2\times \mathrm{O}_2$-symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in $\mathbb{R}^4$ from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in $x_1$-direction, but logarithmically slow in $x_2$-direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in $x_1$-direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.

Classification of ancient noncollapsed flows in $\mathbb{R}^4$

TL;DR

The paper achieves a complete classification of ancient noncollapsed mean curvature flow solutions in , showing they must be among classical shrinkers, 3D bowls/ovals, or members of one-parameter translator/oval families (including HIMW and DH_ovals). A cornerstone is the differential neck theorem, which detects a small yet crucial slope term in mixed bubble-sheet convergence via a differential Merle-Zaag framework, anisotropic barriers, and propagation of smallness. The results imply that any noncompact strictly convex ancient noncollapsed flow is self-similarly translating, ruling out exotic ovals and yielding a canonical neighborhood description for mean-convex flows in . The work closes the classification program in four dimensions by separating bounded from unbounded bubble-sheet scales and delivering a precise translator-ruled asymptotic geometry with strong barrier-based control.

Abstract

In this paper, we classify all noncollapsed singularities of the mean curvature flow in . Specifically, we prove that any ancient noncollapsed solution either is one of the classical historical examples (namely , 2d-bowl, 2d-oval, the rotationally symmetric 3d-bowl, or a cohomogeneity-one 3d-oval), or belongs to the 1-parameter family of -symmetric 3d-translators constructed by Hoffman-Ilmanen-Martin-White, or belongs to the 1-parameter family of -symmetric ancient 3d-ovals constructed by Du-Haslhofer. In light of the five prior papers on the classification program in from our collaborations with Du, Hershkovits, and Choi-Daskalopoulos-Sesum, the major remaining challenge is the case of mixed behaviour, where the convergence to the round bubble-sheet is fast in -direction, but logarithmically slow in -direction. To address this, we prove a differential neck theorem, which allows us to capture the (dauntingly small) slope in -direction. To establish the differential neck theorem, we introduce a slew of new ideas of independent interest, including switch and differential Merle-Zaag dynamics, anisotropic barriers, and propagation of smallness estimates. Applying our differential neck theorem, we show that every noncompact strictly convex solution is selfsimilarly translating, and also rule out exotic ovals.

Paper Structure

This paper contains 22 sections, 60 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. The classification program in $\mathbb{R}^4$
  3. The differential neck theorem and its consequences
  4. Outline of the proofs
  5. Notation and preliminaries
  6. Merle-Zaag type switch dynamics
  7. Elongated cylindrical regions
  8. Anisotropic propagation of smallness
  9. Differential Merle-Zaag analysis
  10. Differential Merle-Zaag alternative
  11. Ruling out the neutral slope mode
  12. Cap distance and consequences
  13. Level set propagation of smallness
  14. Differential neck theorem
  15. Weak differential neck theorem
  16. ...and 7 more sections

Key Result

Theorem 1.1

Any ancient noncollapsed flow in $\mathbb{R}^3$ is, up to scaling and rigid motion, either the static plane $\mathbb{R}^2$, the round shrinking $S^2$, the round shrinking $\mathbb{R}\times S^1$, the rotationally symmetric 2d-bowl, or the rotationally symmetric 2d-oval from White_nature.

Theorems & Definitions (117)

  • Theorem 1.1: Angenent-Daskalopoulos-Sesum, Brendle-Choi
  • Theorem 1.2: classification
  • Corollary 1.3: blowup limits and canonical neighborhoods
  • Theorem 1.4: normal form
  • Theorem 1.5: no wings
  • Theorem 1.6: bubble-sheet ovals
  • Theorem 1.7: translators
  • Theorem 1.10: differential neck theorem
  • Theorem 1.11: no exotic ovals
  • Theorem 1.12: selfsimilarity
  • ...and 107 more