A nonexistence result for wing-like mean curvature flows in ℝ⁴

Kyeongsu Choi; Robert Haslhofer; Or Hershkovits

Paper · May 27, 2021

A nonexistence result for wing-like mean curvature flows in $R^{4}$

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

math.DGWider flowsarXiv:2105.13100v2

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Abstract

Some of the most worrisome potential singularity models for the mean curvature flow of $3$ -dimensional hypersurfaces in $R^{4}$ are noncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to a wedge. In this paper, we rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in $R^{4}$ . Specifically, we prove that for any ancient noncollapsed mean curvature flow $M_{t} = \partial K_{t}$ in $R^{4}$ the blowdown $lim_{λ \to 0} λ \cdot K_{t_{0}}$ is always a point, halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proof we introduce a fine bubble-sheet analysis, which generalizes the fine neck analysis that has played a major role in many recent papers. Our result is also a key first step towards the classification of ancient noncollapsed flows in $R^{4}$ , which we will address in a series of subsequent papers.

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Submitted: May 27, 2021
Last revised: Jun 2, 2023 (v2)
Subjects: math.DG, math.AP
Journal: Geom. Topol. 28 (2024) 3095-3134
DOI: 10.2140/gt.2024.28.3095

Comments: 40 pages

Mean curvature flow Ancient solutions Singularities (Non)collapsing

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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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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

12B.4
Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[1] 12B.4
Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

[2] 12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.

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

A nonexistence result for wing-like mean curvature flows in $R^{4}$

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A nonexistence result for wing-like mean curvature flows in $R^{4}$