Ancient curve shortening flow in the disc with mixed boundary condition

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 ∂ 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 ∂ D orthogonally, and evolves out of the unstable critical arc at t=-∞ and into the stable one at t=+∞. 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∈∂ 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.