§ 2.51 Riedler shrinkers from isoparametric foliationsEDE4E49195FA

Let $M^{n-1}\subset S^ n$ be a closed isoparametric hypersurface with equal principal-curvature multiplicities. Riedler constructs a closed embedded shrinking hypersurface of topology $S^1\times M$ in $\mathbb R^{n+1}$, formed from homothetic copies of leaves of the associated isoparametric foliation. In this entry we retain the cases with $g=3,4,6$ distinct principal curvatures. The $g=1$ case is Angenent's torus; the $g=2$ product-of-spheres case is deliberately omitted from this addition. For equal multiplicity $m$, $n-1=gm$. One example is $S^1\times SO(3)/(\mathbb Z_2\times \mathbb Z_2)\subset \mathbb R^5$. The admissible multiplicities are those of an existing spherical isoparametric foliation; arbitrary $(g,m)$ are not allowed. A shooting argument closes a geodesic in the two-dimensional orbit metric. This is an ODE existence construction, not an elementary parametrization. The resulting hypersurface is compact, complete and proper, and its flow is $\sqrt{-t}\, \Sigma $ for $t<0$. No uniqueness or stability statement is asserted here.

Sources: [Riedler2023].