Classification of noncollapsed translators in ℝ⁴

Kyeongsu Choi; Robert Haslhofer; Or Hershkovits

Paper · May 28, 2021

Classification of noncollapsed translators in $R^{4}$

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

math.DGWider flowsarXiv:2105.13819v2

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Abstract

In this paper, we classify all noncollapsed singularity models for the mean curvature flow of 3-dimensional hypersurfaces in $R^{4}$ or more generally in $4$ -manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $R^{4}$ is either $R \times$ 2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White.

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Submitted: May 28, 2021
Last revised: Jun 2, 2023 (v2)
Subjects: math.DG, math.AP

Comments: 98 pages

Mean curvature flow Self-shrinkers & translators Singularities (Non)collapsing 3-manifolds

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.
12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

[1] 9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[2] 12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

[3] 12A.8
Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing
Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.

Classification of noncollapsed translators in $R^{4}$

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Classification of noncollapsed translators in R4

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Classification of noncollapsed translators in $R^{4}$