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Minimal hypersurfaces in spheres generated by isoparametric foliations

Junqi Lai, Guoxin Wei

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.

Minimal hypersurfaces in spheres generated by isoparametric foliations

Abstract

We investigate the existence of minimal hypersurfaces in that are generated by the isoparametric foliation of a subsphere . 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 . The resulting hypersurfaces have the topological type , extending the known examples of minimal hypertori ( and ) to a broader class of topologies determined by isoparametric structures.
Paper Structure (7 sections, 24 theorems, 94 equations, 2 figures)

This paper contains 7 sections, 24 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of periodic curves
  4. Preparations for proposition \ref{['prop:2025-10-03-1']}
  5. Proof of first item of proposition \ref{['prop:2025-10-03-1']}
  6. Proof of second item of proposition \ref{['prop:2025-10-03-1']}
  7. Proof of third item of proposition \ref{['prop:2025-10-03-1']}

Key Result

Theorem 1.1

For any isoparametric hypersurface $M$ in $\mathbb{S}^n$, there is a closed embedded minimal hypersurface of topological type $S^1 \times M$ in $\mathbb{S}^{n+1}$. This hypersurface is a union of homothetic copies of the leaves of the isoparametric foliation of $\mathbb{S}^n$ associated to $M$.

Figures (2)

  • Figure 1: Types of the profile curves.
  • Figure 2: The existence of the periodic profile curve

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 40 more