Embedded minimal $S^1$-bundles in $\mathbb{S}^4$

We construct infinitely many embedded minimal hypersurfaces of pairwise distinct irreducible topological types in the unit $4$-sphere $\mathbb{S}^4$, which provides a new answer to a problem of Hsiang.

These examples are topologically principal $S^1$-bundles and Seifert fibered manifolds over closed orientable surfaces.

In particular, for any closed orientable surface $\Sigma_{2k-1}$ of odd genus $n=2k-1$, we show that $S^1\times \Sigma_{2k-1}$ admits a minimal embedding into $\mathbb{S}^4$.

The construction is based on the equivariant min-max theory and the suspended (weighted) Hopf action on $\mathbb{S}^4$.