Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere

We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the cur...

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Main Authors: Bryan, Paul, Louie, Janelle
Format: Report
Language:unknown
Published: arXiv 2014
Subjects:
Differential Geometry math.DG
Analysis of PDEs math.AP
FOS Mathematics
53C44 Primary, 35K55, 58J35 Secondary
North Pole
Online Access:https://dx.doi.org/10.48550/arxiv.1408.5523
https://arxiv.org/abs/1408.5523
id ftdatacite:10.48550/arxiv.1408.5523
record_format openpolar
spelling ftdatacite:10.48550/arxiv.1408.5523 2023-05-15T17:39:48+02:00 Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere Bryan, Paul Louie, Janelle 2014 https://dx.doi.org/10.48550/arxiv.1408.5523 https://arxiv.org/abs/1408.5523 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Differential Geometry math.DG Analysis of PDEs math.AP FOS Mathematics 53C44 Primary, 35K55, 58J35 Secondary Preprint Article article CreativeWork 2014 ftdatacite https://doi.org/10.48550/arxiv.1408.5523 2022-04-01T12:49:34Z We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss-Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preserved under the flow. : 16 pages, 1 figure, amsart document class Fixed minor errors in Proposition 2.1 Report North Pole DataCite Metadata Store (German National Library of Science and Technology) North Pole
institution Open Polar
collection DataCite Metadata Store (German National Library of Science and Technology)
op_collection_id ftdatacite
language unknown
topic Differential Geometry math.DG
Analysis of PDEs math.AP
FOS Mathematics
53C44 Primary, 35K55, 58J35 Secondary
spellingShingle Differential Geometry math.DG
Analysis of PDEs math.AP
FOS Mathematics
53C44 Primary, 35K55, 58J35 Secondary
Bryan, Paul
Louie, Janelle
Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
topic_facet Differential Geometry math.DG
Analysis of PDEs math.AP
FOS Mathematics
53C44 Primary, 35K55, 58J35 Secondary
description We prove that the only closed, embedded ancient solutions to the curve shortening flow on $\mathbb{S}^2$ are equators or shrinking circles, starting at an equator at time $t=-\infty$ and collapsing to the north pole at time $t=0$. To obtain the result, we first prove a Harnack inequality for the curve shortening flow on the sphere. Then an application of the Gauss-Bonnet, easily allows us to obtain curvature bounds for ancient solutions leading to backwards smooth convergence to an equator. To complete the proof, we use an Aleksandrov reflection argument to show that maximal symmetry is preserved under the flow. : 16 pages, 1 figure, amsart document class Fixed minor errors in Proposition 2.1
format Report
author Bryan, Paul
Louie, Janelle
author_facet Bryan, Paul
Louie, Janelle
author_sort Bryan, Paul
title Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
title_short Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
title_full Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
title_fullStr Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
title_full_unstemmed Classification of Convex Ancient Solutions to Curve Shortening Flow on the Sphere
title_sort classification of convex ancient solutions to curve shortening flow on the sphere
publisher arXiv
publishDate 2014
url https://dx.doi.org/10.48550/arxiv.1408.5523
https://arxiv.org/abs/1408.5523
geographic North Pole
geographic_facet North Pole
genre North Pole
genre_facet North Pole
op_rights arXiv.org perpetual, non-exclusive license
http://arxiv.org/licenses/nonexclusive-distrib/1.0/
op_doi https://doi.org/10.48550/arxiv.1408.5523
_version_ 1766140572056158208

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