Minimal hypersurfaces in spheres generated by isoparametric foliations

Abstract

We investigate the existence of minimal hypersurfaces in ${\mathbb{S}}^{n+1}$ that are generated by the isoparametric foliation of a subsphere ${\mathbb{S}}^n$. By considering a generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, we reduce the minimal surface equation to an ordinary differential equation. We prove that this construction yields a closed embedded minimal hypersurface for any choice of isoparametric hypersurface $M\subset {\mathbb{S}}^n$. The resulting hypersurfaces have the topological type $S^1\times M$, extending the known examples of minimal hypertori ($S^1\times S^k\times S^k$ and $S^1\times S^k\times S^l$) to a broader class of topologies determined by isoparametric structures.