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

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 $\#^{2g} (S^{1} \times S^{n-1})$.