As on arXiv
Abstract
Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least , while in dimensions and a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions and throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold
Topics Curvature pinching · Shared terms “planarity estimate”, “Baker”, “Naff”
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Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds
Topics Curvature pinching, Singularities · Shared terms “high codimension”, “high curvature”
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Local Convexity and Planarity Estimates for Mean Curvature Flow
Topics Curvature pinching · Shared terms “planarity estimate”
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Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery
Topics Curvature pinching · Same author Stephen Lynch · Shared terms “quadratically pinched”
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Mean Curvature Flow of High Codimension in Complex Projective Space
Topics Curvature pinching, Singularities · Shared terms “high codimension”, “high curvature”
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Classification of quadratically pinched self-shrinkers in higher codimension
Topics Curvature pinching · Shared terms “quadratically pinched”, “Baker”, “Andrews”
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Classification of quadratically pinched translators in higher codimension
Topics Curvature pinching · Shared terms “quadratically pinched”, “stronger pinching”, “pinching range”
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Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere
Topics Curvature pinching, Singularities · Shared terms “high curvature”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12B.6Bounded Curvature at Bounded Distance
The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.