A collapsing ancient solution of mean curvature flow in ℝ³

Theodora Bourni; Mat Langford; Giuseppe Tinaglia

Paper · May 19, 2017

A collapsing ancient solution of mean curvature flow in $R^{3}$

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

math.DGWider flowsarXiv:1705.06981v2

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Abstract

We construct a compact, convex ancient solution of mean curvature flow in $R^{n + 1}$ with $O (1) \times O (n)$ symmetry that lies in a slab of width $π$ . We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O (n)$ -invariant ancient solution that lies in a slab of width $π$ and in no smaller slab.

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Submitted: May 19, 2017
Last revised: Sep 30, 2017 (v2)
Subjects: math.DG

Comments: In this new version, we have made significant revisions to accommodate the following additional material: we derive detailed asymptotics for the constructed solution, we show that this solution is the unique solution amongst compact, convex, rotationally symmetric ancient solutions in a slab, and we generalize the entire process to any dimension

Mean curvature flow Ancient solutions (Non)collapsing

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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) 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) (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.
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.

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

[3] 12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

A collapsing ancient solution of mean curvature flow in $R^{3}$

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A collapsing ancient solution of mean curvature flow in R3

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A collapsing ancient solution of mean curvature flow in $R^{3}$