On uniqueness and nonuniqueness of ancient ovals

Wenkui Du; Robert Haslhofer

Paper · May 28, 2021

On uniqueness and nonuniqueness of ancient ovals

Wenkui Du, Robert Haslhofer

math.DGWider flowsarXiv:2105.13830v2

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Abstract

In this paper, we prove that any nontrivial $SO (k) \times SO (n + 1 - k)$ -symmetric ancient compact noncollapsed solution of the mean curvature flow agrees up to scaling and rigid motion with the $O (k) \times O (n + 1 - k)$ -symmetric ancient ovals constructed by Hershkovits and the second author. This confirms a conjecture by Angenent-Daskalopoulos-Sesum. On the other hand, for every $k \geq 2$ we also construct a $(k - 1)$ -parameter family of uniformly $(k + 1)$ -convex ancient ovals that are only $Z_{2}^{k} \times O (n + 1 - k)$ -symmetric. This gives counterexamples to a conjecture of Daskalopoulos.

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Submitted: May 28, 2021
Last revised: Mar 11, 2022 (v2)
Subjects: math.DG, math.AP

Comments: 36 pages (v2: minor changes in Section 4)

Mean curvature flow Ancient solutions (Non)collapsing Stability & uniqueness

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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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.

On uniqueness and nonuniqueness of ancient ovals

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