The affine shape of a figure 8 under the curve shortening flow
The paper under review advances the theory of geometric flows and deals with the curve shortening flow for closed curves in the Euclidean plane \(\mathbb R^2\).\N\NBy definition, the curve shortening flow is represented by a smooth family of closed curves \(C: S^1\times [0,T)\to \mathbb R^2\) whose evolution in time \(t\) is governed by the equation \(\frac{dC}{dt} = k N\), where \(k\) and \(N\) are the curvature and the unit normal of the evolving curve, respectively.\N\NIt is well-known that if an initial closed curve \(C_0=C(\cdot, 0)\) is embedded, then under the flow it shrinks to a ``round'' point in a finite time \(T\), see [\textit{M. Gage} and \textit{R. S. Hamilton}, J. Differ. Geom. 23, 69--96 (1986; Zbl 0621.53001); \textit{M. A. Grayson}, J. Differ. Geom. 26, 285--314 (1987; Zbl 0667.53001)].\N\NOn the other hand, if \(C_0\) has self-intersections then its evolution under the flow turns out to be more complicated. Particularly, if \(C_0\) is a figure-eight curve, a closed curve with one point of self-intersection, then either of the following two cases can happen: \N(i) if the two lobes of \(C_0\) have different areas, then the curve evolves so that one lobe shrinks to a point before the vanishing time \(T\), the flow expands through this singularity, and then the evolving curve shrinks to a point; \N(ii) if the lobes of \(C_0\) have the same area, then the double point doesn't disappear before the vanishing time \(T\) and the isoperimetric ratio of the evolving curve grows to \(\infty\) as \(t\to T\), see [\textit{M. A. Grayson}, Invent. Math. 96, No. 1, 177--180 (1989; Zbl 0669.53003)]. \NIt is conjectured that in the case (ii) the shrinking to a point holds true as well. The conjecture is confirmed in some particular cases only, but in general the problem remains unsolved.\N\NInspired by the Grayson conjecture, the authors explore the curve shortening flow for a particular family of figure-eight curves. Namely, a figure-eight curve \(C: S^1\to \mathbb R^2\) under consideration is assumed to satisfy the following requirements: \N\begin{itemize}\N\item[(a)] \(C\) is real analytical; \N\item[(b)] the lobes of \(C\) are convex; \N\item[(c)] \(C\) has 4-fold dihedral symmetry. \N\end{itemize}\NMoreover, Cartesian coordinates \((x,y)\) in \(\mathbb R^2\) being specified so that the symmetry axis of \(C\) which intersects \(C\) at three points stands for the coordinate axis \(x\), the curve \(C\) is parameterized by the angle \(\theta\) between the tangent to \(C\) and the coordinate axis \(x\) so that the upper right quarter of \(C\) corresponds to \(-\alpha\leq \theta\leq \frac{\pi}{2}\). \NThen the following additional requirements of \textit{ monotonicity} concerning the curvature \(k(\theta)\) of \(C\) are imposed: \N\begin{itemize}\N\item[(d)] \(\frac{dk}{d\theta}>0\) for all \(-\alpha < \theta < \frac{\pi}{2}\); \N\item[(e)] \(\frac{d^2k}{d\theta^2}\not= 0\) at \(\theta=\frac{\pi}{2}\); \N\item[(f)] the curvature of \(C\), as a function of arc length, does not vanish to second order at the double point.\N\end{itemize}\NEvidently, the family of figure-eight curves satisfying the requirements (a)--(e) is quite large and, in some sense, can be viewed as generic. The most illustrative example is provided by the lemniscate of Bernulli.\N\NThe authors explore the impact of the curve shortening flow on figure-eight curves in question. It is shown that if an initial figure-eight curve satisfies (a)-(e), then the assumptions (a)--(e) are preserved by the flow and the curve shrinks so that a particular suitably rescaled arc of the evolving curve converges to the Grim Reaper Soliton, c.f. [\textit{S. Angenent}, J. Differ. Geom. 33, No. 3, 601--633 (1991; Zbl 0731.53002)].\N\NMoreover, a specific normalization of the curve shortening flow is introduced for clarifying the phenomenon of shrinking for figure-eights curves. Namely, for an arbitrary figure-eight curve, the authors consider the bounding box, the smallest rectangle containing the curve, and then apply to the curve an affine transformation in \(\mathbb R^2\), represented by a positive diagonal matrix, which transform the bounding box to the square \([-1,1]^2\). Consequently, the curve shortening flow is normalized so that the square \([-1,1]^2\) stands as the bounding box of the evolving curve for all \(0\leq t<T\). It is proved that if the initial figure-eight curve satisfies (a)-(e), then under the normalized curve shortening flow it converges (in the Hausdorff metric) to the bowtie, the quadrilateral consisting of four segments connecting cyclically the points $(1,1), (-1,-1), (-1,1), (1,-1)$.
scientific article; zbMATH DE number 7902274