The blowdown of ancient noncollapsed mean curvature flows

Wenkui Du; Robert Haslhofer

Paper · Jun 8, 2021

The blowdown of ancient noncollapsed mean curvature flows

Wenkui Du, Robert Haslhofer

math.DGWider flowsarXiv:2106.04042

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Abstract

In this paper, we consider ancient noncollapsed mean curvature flows $M_{t} = \partial K_{t} \subset R^{n + 1}$ that do not split off a line. It follows from general theory that the blowdown of any time-slice, $lim_{λ \to 0} λ K_{t_{0}}$ , is at most $n - 1$ dimensional. Here, we show that the blowdown is in fact at most $n - 2$ dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly $k$ -convex case, the blowdown is at most $k - 2$ dimensional. This generalizes recent results from Choi-Haslhofer-Hershkovits to higher dimensions, and also has some applications towards the classification problem for singularities in 3-convex mean curvature flow.

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Submitted: Jun 8, 2021
Subjects: math.DG, math.AP

Comments: 22 pages. arXiv admin note: text overlap with arXiv:2105.13100

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.

The blowdown of ancient noncollapsed mean curvature flows

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