§ 5.39 Ancient isoparametric flows in spheres2DFF09123589

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.

For a concrete hypersurface in the unit sphere, take $S^ p(a(t))\times S^ q(b(t))\subset S^{p+q+1}$, $p,q\ge 1$, with $a^2+b^2=1$. If $n=p+q$, the exact evolution is

Here $C\ne 0$ and the time interval is the maximal interval on which both squared radii are positive, extending to $-\infty $. The $C=0$ Clifford leaf is stationary. This formula is intrinsic MCF in the fixed round sphere, not Euclidean shrinkage. The $(p,q)=(1,1)$ illustration is a projection of tori in $S^3\subset \mathbb R^4$.

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].