Classification of convex ancient free-boundary curve-shortening flows in the disc (Q6145248)

This article is concerned with the study of curve-shortening flow in the unit disc. Precisely, it is shown that up to a rotation about the origin and a translation in time, there exists exactly one convex, locally uniformly convex ancient solution to the free-boundary curve-shortening flow in the unit disc. Such a solution has the following properties:\\ (i) It converges for \(t\to 0\) to the point \((0, 1)\),\\ (ii) It converges for \(t\to -\infty\) to the segment line \([-1, 1]\times \{0\}\),\\ (iii) It is invariant under reflection across the \(y\)-axis,\\ (iv) As a graph over the \(x\)-axis, it satisfies \(e^{\lambda^2 t} y(x, t)\to A \cosh(\lambda x)\) uniformly in \(x\) as \(t\to -\infty\), for some \(A>0\), where \(\lambda\) is the solution to \(\lambda \tanh \lambda=1\). The second main result of the article states that there exist no proper rotating solutions for the free-boundary curve-shortening flow in the unit disc.