Which shapes can appear in a curve shortening flow singularity? (Q6633349)

A family of closed curves is given by a differentiable map \(\gamma :[0,T)\times \mathbb{R}\rightarrow \mathbb{R}^{2}\), where \(u\rightarrow \gamma (t,u)\) parametrizes the curve at time \(t\in \lbrack 0,T)\), the curves satisfying \(\gamma (t,u+1)=\gamma (t,u)\) for all \(t,u\).\ The curves are immersed if \(\gamma _{u}(t,u)(=\frac{\partial \gamma }{\partial u})\neq 0\) for all \(t,u\). A family \(\gamma (t,\cdot )\) of curves evolves by curve shortening flow if it satisfies \[\gamma _{t}(t,u)^{\perp }=k(t,u),\] where \( \gamma _{t}^{\perp }\) is the component of \(\gamma _{t}\) that is perpendicular to \(\gamma _{u}\). The map \(\gamma :[0,T)\times \mathbb{R} \rightarrow \mathbb{R}^{2}\) is called a normal parametrization if \(\gamma _{t}\perp \gamma _{u}\) for all \((t,u)\). \N\NThe authors recall that for any initial curve \(\gamma ^{0}\) there is a unique solution to the curve shortening flow starting from \(\gamma ^{0}\). The maximal time interval \([0,T)\) during which the solution \(\gamma \) is defined depends on the initial curve, but it is always finite, the maximal curvature on the curve becoming infinite. \N\NThe authors recall the Gage-Hamilton-Grayson theorem for embedded curves [\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)] and Oaks' theorem for immersed curves [\textit{J. A. Oaks}, Indiana Univ. Math. J. 43, No. 3, 959--981 (1994; Zbl 0835.53048)].\ They describe two examples of curves with self-intersections: the symmetric figure-eight and the cardioid, for which exactly one self-intersection of the solution vanishes into the singularity, and they describe their evolution through curve shortening flow. The paper presents and analyzes different examples of solutions where a singularity absorbs more than one self-intersection. The authors define a loop of a solution to curve shortening flow \(\gamma :(t_{0},t_{1})\times \mathbb{R}\rightarrow \mathbb{R}^{2}\) as a pair of functions \( a,b:(t_{0},t_{1})\rightarrow \mathbb{R}\) such that \(\gamma (t,a(t))=\gamma (t,b(t))\) for all \(t\in (t_{0},t_{1})\), and \(\gamma (t,\cdot )\) is injective on \([a(t),b(t)]\), and the area of a loop. Considering more complicated figures that vanish into a singular point, they define a flat knot in the plane as a closed immersed curve in \(\mathbb{R}^{2}\) that has no self-tangencies. An immersed curve \(\gamma \) has a tangle in \(S\) if \(\gamma \) intersects \(S\) transversely in two distinct points, and if \(\gamma \) has no self-tangencies inside \(S\). The authors finally analyze the cases of \(n\)-loop curves, \(n=3,4\) and \(n\geq 5\), describing with details and through numerical simulations some examples.