Curve-straightening in Riemannian manifolds
type: Item
instance of: scholarly article
title: Curve-straightening in Riemannian manifolds
zbMATH Open document ID: 0653.53032
DOI: 10.1007/BF00127856
author: David A. Singer
author: Joel C. Langer
published in: Annals of Global Analysis and Geometry
publication date: 1987-01-01T00:00:00Z
review text: This paper deals with the bending energy of an immersed curve. In a previous paper [Topology 24, 75-88 (1985; Zbl 0561.53004)] the authors studied elastic curves as critical points of the functional \(\int (k\quad 2+\lambda)ds\) where k is the curvature and \(\lambda\) is a real number. Here they study the possibility of a ``flow of steepest descent'' for this functional including a so-called ``strengthened'' Palais-Smale condition. The particular case of curves in the standard two-sphere is examined in detail, and for certain values of \(\lambda\) the stability of the critical points is analyzed. Here the values \(\lambda =8/7\) and \(\lambda =10/9\) for the sphere of radius 1 play a particular role. From the paper: ``The transition between behavior for small values of \(\lambda\) and large ones is somewhat mysterious at present.''
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 4065855
zbMATH Keywords: wave like elastica
zbMATH Keywords: bending energy
zbMATH Keywords: immersed curve
zbMATH Keywords: elastic curves
zbMATH Keywords: flow of steepest descent
zbMATH Keywords: Palais-Smale condition
Wikidata QID: Q115395251
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
cites work: Q5826059
cites work: The heat equation shrinking convex plane curves
cites work: The total squared curvature of closed curves
cites work: Curves in the Hyperbolic Plane and mean Curvature of Tori in 3-Space
cites work: Curve straightening and a minimax argument for closed elastic curves
cites work: A general approach to Morse theory
description: scientific article; zbMATH DE number 4065855