§ 5.22 Non-self-similar ancient doughnuts1E48CF8C9E56

For each $n\ge 2$, Bourni–Langford–Mramor construct a closed embedded rotational ancient flow with topology $S^1\times S^{n-1}$ in $\mathbb R^{n+1}$ that is not a soliton. It is obtained by perturbing a shrinking doughnut inward and taking an ancient limit. In rescaled coordinates the backward limit is a shrinking doughnut; the finite-time evolution is not a family of homothetic copies of that limit. The source explains that the flow develops a circle of neck-pinch singularities at the forward endpoint. The regular slices are compact, complete and proper. The flow is not mean convex for all ancient times, but becomes mean convex before the final circle of neck pinches. In dimension two their genus is one and they have no ends. For the construction based on the Drugan–Nguyen doughnut, entropy is strictly less than two, by the input entropy bound and monotonicity. No exact entropy value, closed parametrization or uniqueness of the ancient solution is claimed. The figure distinguishes the rescaled backward shrinker profile from the later thin tube around a circle. Its evolving tube outline is a geometric schematic, not a generic torus mislabeled as the Angenent torus or a numerical solution of the ancient PDE.

Sources: [BourniLangfordMramor2020].