As on arXiv
Abstract
In a recent breakthrough, Bamler-Lai proved the mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is – up to splitting Euclidean factors – either a translating bowl, or an ancient oval, or a translating oval-bowl. In this paper, we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, the present paper also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the approach from our prior work joint with Hershkovits (Acta '22) and Hershkovits-White (Inventiones '22).
Comments: 13 pages
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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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Classification of ancient noncollapsed flows in
Topics Ancient solutions, Singularities · Same authors Kyeongsu Choi, Robert Haslhofer · Shared terms “Hershkovits”, “White”, “Specifically”
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Free boundary flow through cylindrical singularities
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Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
Topics Ancient solutions, Singularities · Shared terms “ancient cylindrical”, “neighborhood conjecture”, “convex neighborhood”
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Enhanced profile estimates for ovals and translators
Topics Ancient solutions · Same authors Kyeongsu Choi, Robert Haslhofer · Shared terms “ancient ovals”
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Revisiting generic mean curvature flow in
Topics Ancient solutions, Singularities · Same author Kyeongsu Choi · Shared terms “Bamler”
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Topics Ancient solutions, Singularities · Same author Robert Haslhofer · Shared terms “Hershkovits”, “oval bowl”, “ancient ovals”
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Classification of ancient ovals in higher dimensional mean curvature flow
Topics Ancient solutions · Shared terms “ancient ovals”, “Bamler Lai”, “neighborhood conjecture”
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Classification of bubble-sheet ovals in
Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “ancient ovals”, “Hershkovits”
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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.
In the guidebook
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12B.1κ-Solutions
Ancient, nonnegatively curved, noncollapsed: the models for singularities.