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

Abstract

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 ${\mathbb{S}}^4$.

34 pages, 4 figures. A few details added and some typos corrected. Comments are welcome!