Classification of solitons for the affine curvature flow
type: Item
instance of: scholarly article
title: Classification of solitons for the affine curvature flow
zbMATH Open document ID: 0941.53039
author: L. L. de Lima
author: José Fabio B. Montenegro
published in: Communications in Analysis and Geometry
publication date: 2000-08-03T00:00:00Z
review text: Let \(X= X(t,u): [0,T)\times I\to \mathbb{R}^2\) be a one-parameter family of closed curves. It is a solution of the affine curvature flow if (1) \(X_t= -kN\), where \(k\) denotes the affine curvature and \(N\) the affine normal of \(X(t,.)\). The authors are interested in classifying the homothetic solutions (``solitons'') of this flow. Their approach is inspired by [\textit{U. Abresch} and \textit{J. Langer}, J. Differ. Geom. 23, 175-196 (1986; Zbl 0592.53002)] where said problem is treated for the curve shortening flow. The authors first renormalize the flow by adding a tangent component \(gT\) to (1) so that each \(X(t,.)\) has constant affine speed. This enables them to derive the evolution equation for the renormalized curvature \(\kappa= \xi^2k\), where \(\xi\) denotes the affine velocity. The resulting fourth-order nonlinear parabolic equation for \(\kappa\) can then be reduced by virtue of the ``soliton condition'' to the second-order ordinary differential equation (2) \(\ddot\kappa+ \kappa^2-1=0\), whose periodic solutions can be expressed in terms of elliptic functions. After some pleasant jugglery utilizing the method of adjoint orbits, the authors are able to write down explicitly the homothetic solutions of (1) in terms of \(\kappa\), leading to the following end result. Theorem. Let \(X\) be a homothetic solution of (1). Then either (i) \(X\) is an ellipse (corresponding to the constant solution \(\kappa= 1\) in (2)), or (ii) for some \((p,q)\in \mathbb{N}\times \mathbb{N}\) with \({1\over 2}<{p\over q}< {1\over\sqrt 2}\), \(X= X_{p,q}\) is a closed curve with winding number \(p\) and with \(2q\) sextactic points. The authors remark at the end that their method can straightforwardly be carried over to the so-called affine-invariant flow \(X_t= N\) which was studied in a series of papers by G. Sapiro, A. Tannenbaum and coauthors; in this case, it turns out that the only homothetic solutions are ellipses.
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 1384293
zbMATH Keywords: affine curvature flow
zbMATH Keywords: curve shortening flow
zbMATH Keywords: method of adjoint orbits
reviewed by: Aleksandar Perović
MaRDI profile type: Publication
full work available at URL: cag.1999.v7.n4.a3
OpenAlex ID: W2740672234
DOI: 10.4310/CAG.1999.V7.N4.A3
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description: scientific article; zbMATH DE number 1384293