Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large first Betti number on spheres

TL;DR

This paper proves regularity results for cohomogeneity two equivariant minimization problems and constructs minimal hypersurfaces on spheres with large Betti numbers, advancing min-max theory and geometric analysis.

Contribution

It develops a cohomogeneity two equivariant min-max theory for minimal hypersurfaces and constructs examples with arbitrarily large Betti numbers on spheres.

Findings

Established regularity of cohomogeneity two equivariant isotopy minimization problems.

Constructed minimal hypersurfaces with large Betti numbers on spheres.

Showed convergence of these hypersurfaces to unions involving Clifford hypersurfaces.

Abstract

We prove the regularity of cohomogeneity two equivariant isotopy minimization problems. Based on this, we develop cohomogeneity two equivariant min-max theory for minimal hypersurfaces proposed by Pitts and Rubinstein in 1988. As an application, for $g \ge 1$ and $4 \le n+1 \le 7$, we construct minimal hypersurfaces $\Sigma_{g}^{n}$ on round spheres $\mathbb{S}^{n+1}$ with $(SO(n-1) \times \mathbb{D}_{g+1})$-symmetry. For sufficiently large $g$, $\Sigma_{g}^{n}$ is a sequence of minimal hypersurfaces with arbitrarily large Betti numbers of topological type $\#^{2g} (S^{1} \times S^{n-1})$ or $\#^{2g+2} (S^{1} \times S^{n-1})$, which converges to a union of $\mathbb{S}^{n}$ and a Clifford hypersurface $\sqrt{\frac{1}{n}}\mathbb{S}^{1} \times \sqrt{\frac{n-1}{n}} \mathbb{S}^{n-1}$ or $\sqrt{\frac{2}{n}}\mathbb{S}^{2} \times \sqrt{\frac{n-2}{n}} \mathbb{S}^{n-2}$. In particular, for dimensions $5$ and $6$, $\Sigma_{g}^{n}$ has a topological type $\#^{2g} (S^{1} \times S^{n-1})$.