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A
nonexistence result for wing-like mean curvature flows in
$\mathbb{R}^4$
Kyeongsu Choi, Robert Haslhofer and Or Hershkovits |
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Geometry & Topology 28 (2024) 3095–3134
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Abstract |
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Some of the most worrisome potential singularity models for the mean curvature flow of three-dimensional hypersurfaces in are noncollapsed wing-like flows, ie noncollapsed flows that are asymptotic to a wedge. We rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in . Specifically, we prove that for any ancient noncollapsed mean curvature flow in the blowdown is always a point, halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proof we introduce a fine bubble-sheet analysis, which generalizes the fine neck analysis that has played a major role in many recent papers. Our result is also a key first step towards the classification of ancient noncollapsed flows in , which we will address in a series of subsequent papers. |
Keywords
mean curvature flow, singularities, ancient solutions
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Mathematical Subject Classification
Primary: 53E10
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References |
Publication
Received: 11 August 2021
Revised: 2 May 2023
Accepted: 2 June 2023
Published: 25 November 2024
Proposed: Tobias H Colding
Seconded: David Fisher, Bruce Kleiner
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Authors |
| Kyeongsu Choi | |
| School of Mathematics Korea Institute for Advanced Study Seoul South Korea |
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| Robert Haslhofer | |
| Department of Mathematics University of Toronto Toronto, ON Canada |
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| Or Hershkovits | |
| Institute of Mathematics Hebrew University Jerusalem Israel |
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| © 2024 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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