§ 5.7 Immersed ancient curves from grim-reaper chainsF7D04537B40F

Angenent–You construct compact and noncompact immersed ancient curve-shortening flows from ordered chains of grim reapers. Consecutive horizontal asymptotes $y=a_{k-1},a_ k$, with distinct adjacent heights, determine the grim speed $v_ k=\pi /|a_ k-a_{k-1}|$ and the direction is chosen as in their formula (3). The complete cyclic closing and intersection requirements in Sections 2–4 are essential; arbitrary superposition is not a solution. The compact case has a closing grim arc and parameter domain $S^1$. The noncompact construction has parameter domain $\mathbb R$, two ends asymptotic to the first and last horizontal lines, and the parity condition in the source. The construction includes locally convex immersed curves and curves that are neither embedded nor locally convex. These are distinct from the previously recorded embedded eternal chain examples. Their time interval is $(-\infty ,T)$ for some endpoint $T$, with no general eternal continuation claim for the compact case. The figure shows the limiting grim pieces and connections at several increasingly late schematic stages. The finite-time gluing correction is not computed, and crossings in the intended planar immersed example are distinguished from artifacts of a higher-dimensional projection.

Sources: [AY18].