Isoperimetric inequalities for special classes of curves
type: Item
instance of: scholarly article
title: Isoperimetric inequalities for special classes of curves
zbMATH Open document ID: 1218.53005
author: Bernd Süssmann
published in: Differential Geometry and its Applications
publication date: 2011-04-19T00:00:00Z
review text: The curve shortening problem is described by the partial differential equation \[ X_t=kN \tag{1} \] for a closed initial curve \(C(0)\) of class \(C^2\), where \(X\) is a parametrization of the evolving family of curves \(C(t)\), \(N\) is the inward unit normal vector, \(k\) is the curvature, and the subscript \(t\) denotes the partial derivative at time \(t\). Solutions of \((1)\) which do not change their shape during the evolution, are called homothetic. In the paper under review, the classical Banchoff-Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is sharpened for homothetic or Abresch-Langer solutions of curve shortening. For a larger class of curves and for rotationally symmetric curves, further isoperimetric inequalities containing the rotation number and the winding number are presented.
reviewed by: Valeriĭ V. Volchkov
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 5880266
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: Banchoff-Pohl inequality
zbMATH Keywords: winding number
zbMATH Keywords: curve shortening
zbMATH Keywords: homothetic solutions
Wikidata QID: Q115357011
MaRDI profile type: Publication
full work available at URL: j.difgeo.2010.12.005
OpenAlex ID: W1993356098
cites work: The normalized curve shortening flow and homothetic solutions
cites work: A generalization of the isoperimetric inequality
cites work: Q3788657
cites work: A stable manifold theorem for the curve shortening equation
cites work: Curve shortening flow and the Banchoff-Pohl inequality on surfaces of nonpositive curvature
DOI: 10.1016/J.DIFGEO.2010.12.005
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description: scientific article; zbMATH DE number 5880266