§ 2.47 Park bi-rotational shrinking ends51C4E786B6C8

For $2\le k\le n-1$, Park's 2026 preprint constructs bi-rotational shrinking pieces in $\mathbb R^{n+1}$ using the $O(k)\times O(n-k+1)$ action. The trumpet-type domains are $[0,\infty )\times S^{k-1}\times S^{n-k}$; the disk-type pieces have topology $S^{k-1}\times D^{n-k+1}$. The boundary and the allowable orbit-space regions in Theorems 1.1–1.3 are essential. They are not arbitrary globally complete embedded shrinkers. The first ambient dimension is four. The orbit-space schematic displays a truncated meridian, its boundary point and the two sphere radii; it does not certify the global continuation or a closing condition. Under the normalization $\mathbf H=-X^\perp /2$, the end evolves as $\sqrt{-t}\, \Sigma $ for $t<0$, including motion of its inner boundary. No boundary condition for a separate fixed-barrier MCF problem is imposed. No global entropy or uniqueness conclusion beyond the cited end problem is recorded.

Sources: [Park2026].