Minimal hypersurfaces in spheres generated by isoparametric foliations

TL;DR

This paper constructs new minimal hypersurfaces in spheres using isoparametric foliations, extending known examples to broader topologies by reducing the problem to an ordinary differential equation.

Contribution

It introduces a generalized rotational ansatz based on isoparametric foliations, producing a new class of embedded minimal hypersurfaces with diverse topologies.

Findings

Constructed closed embedded minimal hypersurfaces in spheres.

Extended known minimal hypertori to broader topologies.

Reduced the minimal surface equation to an ODE using a new ansatz.

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.