Classification of solitons for the affine curvature flow
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.
scientific article; zbMATH DE number 1384293
scientific article