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A comparison theorem for the isoperimetric profile under curve-shortening flow

Andrews, Benjamin; Bryan, Paul

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We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve-shortening flow: if the isoperimetric profile of the region enclosed by the initial curve is greater than that of some "model" convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the "paperclip" solution as the model region to deduce sharp...[Show more]

dc.contributor.authorAndrews, Benjamin
dc.contributor.authorBryan, Paul
dc.date.accessioned2015-12-10T22:20:37Z
dc.identifier.issn1019-8385
dc.identifier.urihttp://hdl.handle.net/1885/52008
dc.description.abstractWe prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve-shortening flow: if the isoperimetric profile of the region enclosed by the initial curve is greater than that of some "model" convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the "paperclip" solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve-shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates or classification of self-similar solutions.
dc.publisherInternational Press
dc.sourceCommunications in Analysis and Geometry
dc.titleA comparison theorem for the isoperimetric profile under curve-shortening flow
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolume19
dc.date.issued2011
local.identifier.absfor010199 - Pure Mathematics not elsewhere classified
local.identifier.ariespublicationf5625xPUB237
local.type.statusPublished Version
local.contributor.affiliationAndrews, Benjamin, College of Physical and Mathematical Sciences, ANU
local.contributor.affiliationBryan, Paul, College of Physical and Mathematical Sciences, ANU
local.description.embargo2037-12-31
local.bibliographicCitation.issue3
local.bibliographicCitation.startpage503
local.bibliographicCitation.lastpage539
dc.date.updated2015-12-09T08:48:38Z
local.identifier.scopusID2-s2.0-80054006991
local.identifier.thomsonID000296081800003
CollectionsANU Research Publications

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