Symmetrized curve-straightening.
type: Item
instance of: scholarly article
title: Symmetrized curve-straightening.
zbMATH Open document ID: 1035.53093
DOI: 10.1016/S0926-2245(02)00144-4
author: Anders Linnér
published in: Differential Geometry and its Applications
publication date: 2003-08-25T00:00:00Z
review text: The curve-straightening flow with \(\gamma\geq 0\) is the negative gradient flow associated with the modified total squared curvature functional: \(\nu\to \int_\gamma{1\over 2} k^2+\gamma \,ds\) on the space \(X\) of curves, the larger possible Sobolev space on which the modified total squared curvature functional is defined with the simplest standard metric. The author's choice facilitates the computation of the gradient and its projection onto tangent spaces in the case of constraints. It also simplifies several of the analytical arguments required when verifying the Palais-Smale condition. In this context, the purpose of this paper is to answer questions concerning the failure of the traditional curve-straightening flow to preserve certain symmetries of reflection. The main result is the proof of the existence of such curve-straightening flows. This investigation will also explain the phenomena of ``spinning'' as observed in several examples, in the traditional flow. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 1970794
zbMATH Keywords: Curve-straightening
zbMATH Keywords: Gradient flow
zbMATH Keywords: Steepest descent
zbMATH Keywords: Symmetry
reviewed by: Liliana Răileanu
MaRDI profile type: Publication
Wikidata QID: Q115338111
cites work: Manifolds, tensor analysis, and applications.
cites work: Some Properties of the Curve Straightening Flow in the Plane
cites work: Curve-straightening and the Palais-Smale condition
full work available at URL: s0926-2245(02)00144-4
OpenAlex ID: W2086563939
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description: scientific article; zbMATH DE number 1970794