§ 5.32 Ancient isoparametric flows in Euclidean spaceF1B0B077B481

Liu–Terng prove that the spherical and Euclidean mean curvature flows starting at a compact isoparametric submanifold $M^ n\subset S^{n+k-1}$ extend for all backward time. Isoparametric means flat normal bundle and constant principal curvatures along parallel normal fields. The nonminimal leaves give nonstationary spherical flows and nonhomothetic Euclidean flows. They approach the unique parallel minimal isoparametric leaf backward in the respective normalization, and collapse forward to a focal submanifold.

If $f(s,\cdot )$ is the spherical flow, the Euclidean flow is

A concrete exact case is the product $S^ p(\sqrt{A-2pt})\times S^ q(\sqrt{B-2qt})\subset \mathbb R^{p+q+2}$, for $t<\min (A/(2p),B/(2q))$, with $A/p\ne B/q$. This gives a nonhomothetic ancient flow; the equal-extinction case is already a spherical shrinker. The $(p,q)=(1,1)$ figure projects the exact product tori from four-space, retaining both radii separately.

Smooth compact slices are embedded, complete and proper, and their topology remains that of the initial regular leaf before the focal event. The general isoparametric families also include flag-manifold examples. The explicit product picture illustrates one case, not all such leaves.

Sources: [LiuTerng2020].