Paper · Aug 30, 2022

A nonexistence result for rotating mean curvature flows in $R^{4}$

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Abstract

Some worrisome potential singularity models for the mean curvature flow are rotating ancient flows, i.e. ancient flows whose tangent flow at $- \infty$ is a cylinder $R^{k} \times S^{n - k}$ and that are rotating within the $R^{k}$ -factor. We note that while the $R^{k}$ -factor, i.e. the axis of the cylinder, is unique by the fundamental work of Colding-Minicozzi, the uniqueness of tangent flows by itself does not provide any information about rotations within the $R^{k}$ -factor. In the present paper, we rule out rotating ancient flows among all ancient noncollapsed flows in $R^{4}$ .

Comments: 12 pages

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

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. 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 usually cannot run forever: curvature blows up in finite time, 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.

A nonexistence result for rotating mean curvature flows in $R^{4}$

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A nonexistence result for rotating mean curvature flows in R4

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A nonexistence result for rotating mean curvature flows in $R^{4}$