Ancient curve shortening flow in the disc with mixed boundary condition

Abstract

Given any non-central interior point $o$ of the unit disc $D$, the diameter $L$ through $o$ is the union of two linear arcs emanating from $o$ which meet $\partial D$ orthogonally, the shorter of them stable and the longer unstable (under these boundary conditions). In each of the two half discs bounded by $L$, we construct a convex eternal solution to curve shortening flow which fixes $o$ and meets $\partial D$ orthogonally, and evolves out of the unstable critical arc at $t=-\infty$ and into the stable one at $t=+\infty$. We then prove that these two (congruent) solutions are the only non-flat convex ancient solutions to the curve shortening flow satisfying the specified boundary conditions. We obtain analogous conclusions in the "degenerate" case $o\in\partial D$ as well, although in this case the solution contracts to the point $o$ at a finite time with asymptotic shape that of a half Grim Reaper, thus providing an interesting example for which an embedded flow develops a collapsing singularity.