Cohomogeneity Two Equivariant Isotopy Minimization

• The paper establishes the smooth existence of SO(n–1)×D₍g+1₎-equivariant minimal hypersurfaces in spheres using isotopy minimization and min–max theory.
• It employs the Hsiang–Lawson metric to reduce the problem to three dimensions, ensuring regularity and control of geometric measure up to dimension 7.
• The construction produces high-genus surfaces with arbitrarily large first Betti numbers and prescribed topology, offering a systematic method for generating complex minimal hypersurfaces.

Cohomogeneity two equivariant isotopy minimization consists of the regularity theory, min–max methodology, and geometric/topological consequences for minimal hypersurface construction under cohomogeneity-two group actions, specifically in round spheres of dimension $4 \le n+1 \le 7$. The fundamental results establish the existence and smoothness of minimizing $SO(n-1) \times \mathbb{D}_{g+1}$-equivariant hypersurfaces with arbitrarily large first Betti numbers, via both isotopy minimization and cohomogeneity-two min–max theory (Ko, 13 Dec 2025).

1. Cohomogeneity-Two Symmetry and Orbit Reduction

Let $M = S^{n+1}$, acted on by $G = SO(n-1)\times \mathbb{D}_{g+1}$, where $SO(n-1)$ rotates the first $n-1$ coordinates and $\mathbb{D}_{g+1}$ (the dihedral group of order $2(g+1)$) acts dihedrally on the final two coordinates. The principal $SO(n-1)$-orbits are $(n-2)$-spheres, so the orbit space $M' := M/SO(n-1)$ is homeomorphic to the closed unit ball $\mathbb{B}^3$. The boundary $\partial M'$ corresponds to the collapsed $(n-1)$-sphere where $x_1^2+\cdots+x_{n-1}^2 = 0$. To induce correspondence between equivariant minimal hypersurfaces in $(M,g)$ and minimal surfaces in $M'$, one constructs the Hsiang–Lawson metric:

$$g' = \bigl(\mathrm{Vol}(\pi^{-1}(p))\bigr)^{2/(n-1)}\,\bar g, \qquad p \in M',$$

where $\bar g$ is the quotient metric. Minimal $SO(n-1)$-equivariant hypersurfaces in $(M,g)$ thus correspond to minimal surfaces in $(M',g')$, meeting $\partial M'$ orthogonally at regular boundary points.

2. Equivariant Isotopy-Minimization Problem

Given an admissible open set $U' \subset M'$ (homeomorphic to a $3$-ball, meeting $\partial M'$ “nicely”), define the isotopy family

$$\mathcal{I}(U') = \bigl\{ \varphi : [0,1] \times M' \to M' \mid \varphi(0, \cdot) = \mathrm{Id},\, \varphi(t)|_{M' \setminus U'} = \mathrm{Id},\, \varphi(t) \text{ diffeomorphism} \bigr\}.$$

For a $SO(n-1)$-equivariant hypersurface $\Sigma = \pi^{-1}(\Sigma')$, the isotopy minimization problem is

$$I = \inf_{\varphi \in \mathcal{I}(U')} \mathcal{H}^n \bigl( \pi^{-1}(\varphi(1, \Sigma') \cap M') \bigr),$$

with $\{\Sigma'_i = \varphi^i(1, \Sigma')\}$ a minimizing sequence. The problem reduces via Hsiang–Lawson to area minimization over $(M',g')$ under isotopies supported in $U'$.

3. Regularity Theory for Minimizers

The central regularity result (for $4 \leq n + 1 \leq 7$) asserts: if $\Sigma'_i$ is a minimizing sequence meeting $\partial M'$ transversely, and $\Sigma_i = \pi^{-1}(\Sigma'_i)$ converge as varifolds to $V$, then:

• $V = \Gamma$ is a smooth, embedded, $SO(n-1)$-equivariant minimal hypersurface inside $\pi^{-1}(U')$, with boundary on $\pi^{-1}(\partial U')$.
• On $M \setminus U = \pi^{-1}(M' \setminus U')$, one has $$V|_{M \setminus U} = \pi^{-1}(\Sigma')|_{M \setminus U}$$.
• $\Gamma$ is $SO(n-1)$-stable under normal deformations induced by the isotopy family.

Regularity is achieved by reducing the ambient problem to three dimensions via Hsiang–Lawson, applying Meeks–Simon–Yau $\gamma$-reduction and topological surgery (neck-cutting), employing replacement tricks to isolate minimal disks where area is lower, controlling tangent cone density via Simons' stable minimal cone classification, and excluding singularities through Allard’s and Schoen–Simon’s regularity up to dimension $7$.

4. Equivariant Min–Max Theory

Building on the Pitts–Rubinstein framework, one constructs a one-parameter family (“sweepout”) of $G_f$-invariant surfaces $\{\Lambda_t\}$ in $M' = \mathbb{B}^3$, each of controlled genus. The min–max width is

$$W = \inf_{\{\Lambda_t\} \in \Pi} \sup_{t \in [0,1]} \mathcal{H}^n \bigl( \pi^{-1}(\Lambda_t) \bigr),$$

where $\Pi$ is the $G_f$-saturated class under equivariant isotopies. A pull-tight argument produces a min–max sequence whose varifold limit $V$ is $G$-stationary and almost–minimizing in $G$-annuli. Equivariant local replacements and the regularity theorem ensure that

$V = \sum_{i=1}^k n_i\,\Gamma_i,$

with each $\Gamma_i$ smooth, embedded, pairwise-disjoint, $G$-equivariant minimal hypersurface, and the projected sum of genera satisfies

$\sum_{i=1}^k \genus(\pi(\Gamma_i)) \leq g.$

5. Construction of Minimal Hypersurfaces with Large First Betti Number

Application to $S^{n+1}$ yields, for $g \ge 1$, minimal hypersurfaces $\Sigma_g^n$ exhibiting $(SO(n-1) \times \mathbb{D}_{g+1})$ symmetry, arbitrarily large first Betti numbers, and prescribed topology:

• By parametrizing the Clifford family $\{S_t^2 : |x'|^2 = t\} \subset \mathbb{B}^3$, desingularizing with equatorial disks, and imposing $\mathbb{D}_{g+1}$-symmetry, one constructs sweepouts of genus $g$.
• Min–max area estimates show $W > \mathcal{H}^n(\mathbb{S}^n)$, excluding collapse to equatorial spheres.
• The genus and topology are proved via Simon’s lifting lemma and Euler–Hurwitz enumerations: $\Sigma_g^n$ is diffeomorphic to $\#^{2g}(S^1 \times S^{n-1})$ or, for $n=3,6$, to $\#^{2g+2}(S^1 \times S^{n-1})$.
• In $n+1=5,6$, a Willmore-type area dichotomy ensures only the $S^{n-2} \times S^2$ cross-section, yielding exactly $\#^{2g}(S^1 \times S^{n-1})$.
• As $g \to \infty$, the limit is

$$\lim_{g \to \infty} \Sigma_g = S^n \, \cup\, \sqrt{\frac{2}{n}}\, \mathbb{S}^2 \times \sqrt{\frac{n-2}{n}}\, \mathbb{S}^{n-2},$$

i.e., union of an equator and the corresponding Clifford product.

• Betti number growth is linear: $b_1(\Sigma_g) = 2g$ (or $2g+2$), and Savo’s Morse index estimate gives

$$\text{Index}(\Sigma_g) \gtrsim \frac{4g}{n(n+1)} + n + 2,$$

with both index and topology tending to infinity.

6. Dimension Restrictions and Corollaries

All regularity and embeddedness results hold strictly for $n+1 \le 7$. In higher ambient dimensions, singular sets are expected to have at worst codimension $7$. The genus bound of sweepouts is sharp; for large $g$, the Betti number approximation $b_1(\Sigma_g) \approx 2g$ is valid. The construction extends to equivariant Almgren–Pitts multi-parameter families, guaranteeing an infinite sequence with distinct minimal hypersurface topologies under appropriate symmetries.

7. Synthesis and Foundations

The theory fuses a cohomogeneity-two equivariant extension of the classical embedded disk minimization theorem (Almgren–Simon, Meeks–Simon–Yau) with the Pitts–Rubinstein min–max methodology. For each round sphere $S^{n+1}$, $4 \le n+1 \le 7$, it provides an infinite sequence of embedded minimal hypersurfaces, symmetry $SO(n-1) \times \mathbb{D}_{g+1}$, topologies $\#^{2g}(S^1 \times S^{n-1})$ (and exceptions in special dimensions), with first Betti numbers diverging and degeneration towards equatorial/Clifford configurations as $g$ increases (Ko, 13 Dec 2025).

A plausible implication is the systematic generation of high-genus minimal hypersurfaces in spheres, with precise control over symmetry and topology, extending via equivariant min–max theory to broad classes of symmetric spaces. This suggests promising avenues for Almgren–Pitts multi-parameter variants and further analysis of singularities in higher codimension.