Cohomogeneity Two Equivariant Min-Max Theory

• The paper establishes a framework that combines variational min-max methods with equivariant symmetry reduction to construct minimal hypersurfaces in manifolds with cohomogeneity two actions.
• Methodologies include equivariant sweepouts and isotopy saturations that yield smooth, embedded minimal hypersurfaces with controlled singularities and topological constraints, as seen in classical Lawson examples.
• Applications extend to generating infinite families of minimal hypersurfaces with Weyl law asymptotics, ensuring dense distributions in symmetric Riemannian settings.

Cohomogeneity two equivariant min-max theory is a framework for constructing and analyzing minimal hypersurfaces in Riemannian manifolds with high symmetry, specifically those endowed with an isometric action by a compact Lie group whose principal orbits have codimension two. This theory combines variational min-max principles with equivariant geometric analysis, enabling the production of both classical and new families of embedded minimal hypersurfaces—often with prescribed topological and symmetry constraints—via reduction to a lower-dimensional, often orbifold-structured, orbit space. It has deep connections to the works of Pitts, Rubinstein, and subsequent developments by Hsiang–Lawson, Almgren–Pitts, and others (Ketover, 2016, Wang, 2023, Ko, 13 Dec 2025).

1. Symmetry Reduction and Cohomogeneity Two Group Actions

A manifold $(M^{n+1},g)$ with a smooth, isometric action by a compact Lie group $G$ is said to have cohomogeneity two if the generic (principal) orbits are of codimension two in $M$. Equivalently, the orbit space $Q = M^{\mathrm{prin}}/G$ is a two-dimensional (Riemannian) orbifold with possible lower-dimensional singular strata corresponding to nonprincipal orbits. In practice, $G$ often splits as $G = G_c \times G_f$, where $G_c$ is a connected Lie group and $G_f$ is a finite group generating isolated singular loci (Ko, 13 Dec 2025, Wang, 2023).

For round spheres, a quintessential example is $M = \mathbb{S}^{n+1}$, $G_c = SO(n-1)$ acting on the first $n-1$ coordinates, and $G_f = D_{g+1}$ (the dihedral group), acting on the remaining coordinates. Here, $M/G_c$ is homeomorphic to a $3$-ball, with further quotient by $G_f$ yielding a $2$-dimensional orbifold with stratified singular set. The orbit structure under cohomogeneity two symmetry is essential: the singular set in the orbit space forms a trivalent graph (in dimension $3$) or a stratified submanifold encoding the locations and types of exceptional orbits (Ketover, 2016).

2. Equivariant Sweepouts and Min-Max Formulation

The core variational object is a $G$-equivariant $k$-parameter sweepout, defined as a continuous map

$\Phi : X \to C(M),$

where $X$ is a compact $k$-manifold with boundary, and for each $t \in \mathrm{Int}\,X$, the surface $\Sigma_t := \Phi(t)$ is a smooth, closed, embedded, $G$-invariant hypersurface varying smoothly in $t$. On the topological boundary, the sweepout degenerates in a controlled manner to lower-dimensional $G$-invariant graphs, intersecting the singular set in a manner compatible with the stratification (orthogonally on 1-dimensional strata, meeting 0-dimensional strata on $\partial X$ only) (Ketover, 2016, Ko, 13 Dec 2025).

Given a sweepout $\Phi$, its $G$-equivariant isotopy-saturation $\Pi$ consists of all sweepouts $G$-homotopic to $\Phi$ through families preserving boundary and symmetry conditions. The $G$-width (variational min-max invariant) is then

$$W_G(\Pi) = \inf_{\Psi\in \Pi}\sup_{x\in X} \mathbf{M}(\llbracket \Psi(x)\rrbracket),$$

where $\mathbf{M}$ denotes the mass/area functional (Ketover, 2016, Ko, 13 Dec 2025).

Equivariant min-max sequences constructed in this manner yield subsequential varifold limits that are integer linear combinations of embedded, smooth, $G$-invariant minimal hypersurfaces, with sharp genus-with-multiplicity bounds, intersection properties with singular loci, and controlled neck-pinch degenerations.

3. Analytic Framework: Regularity, Replacement, and Stability

A central analytic pillar is the regularity theory for $G$-stationary and $G$-almost-minimizing varifolds. Key results are:

• Symmetric Criticality: Any $G$-stationary varifold is stationary for all variations, not just those preserving the group action (Ketover, 2016).
• Equivariant Stability Implies Full Stability: For minimal hypersurfaces meeting singular arcs orthogonally, $G$-equivariant stability upscales, permitting the use of standard curvature estimates (Schoen's) to achieve $C^{1,\alpha}$ regularity away from the singular set (Ko, 13 Dec 2025, Ketover, 2016).
• Almost-Minimizing and Replacement: In small $G$-invariant annuli (pullbacks from $M/G$), the min-max sequence is almost-minimizing. The existence of equivariant replacements—area-minimizers among $G$-equivariant competitors—ensures smoothness and rectifiability up to codimension 7 singularities (which by dimension hypothesis do not occur for $4 \leq n+1 \leq 7$) (Ko, 13 Dec 2025).

An important technical feature is that in the $G$-equivariant setting, all local variational procedures (monotonicity, compactness, $\epsilon$-regularity, cut-and-paste minimization) are invoked using $G$-invariant vector fields and isotopies. The intersection behavior of minimal hypersurfaces with the singular axes is tightly constrained: intersections are orthogonal except possibly along $\mathbb{Z}_2$ axes, where multiplicity constraints emerge (Ketover, 2016, Ko, 13 Dec 2025).

4. Examples, Applications, and Classical Recoveries

A wide array of classical minimal surface examples in $\mathbb{S}^3$ and higher spheres are rederived and classified in this framework through their symmetry group and the action on the fundamental domain in the orbit space. Representative cases include:

• Lawson Surfaces: For $G = \mathbb{Z}_{n+1} \times \mathbb{Z}_{m+1}$ acting on the Hopf coordinates of $\mathbb{S}^3$, equivariant min-max recovers Lawson tori of genus $nm$, with $W_G$ equal to their area (Ketover, 2016).
• Choe–Soret and Platonic Examples: Dihedral or platonic actions yield explicit constructions of higher-genus minimal surfaces and recover the Karcher–Pinkall–Sterling surfaces.
• Infinite Families via Doubling and Desingularization: Using the Hopf fibration and pullbacks of geodesic nets in $\mathbb{S}^2$, one constructs infinite families of minimal surfaces (doublings, desingularizations), whose area and genus growth are sharply estimated, and which converge (as varifolds) to multiples of singular stationary varifolds (Ketover, 2016).
• Large Betti Number Hypersurfaces in Spheres: For $G = SO(n-1)\times D_{g+1}$ acting on $\mathbb{S}^{n+1}$, a sweepout by genus-$g$ surfaces leads to minimal hypersurfaces of topological type $\#^{2g}(S^1\times S^{n-1})$ whose first Betti number can be made arbitrarily large, converging to unions of the totally geodesic equator and Clifford hypersurfaces as $g\to\infty$ (Ko, 13 Dec 2025).

5. Weyl Law and Equivariant Volume Spectrum

In cohomogeneity two, the orbit space $(Q, g_Q)$ is a two-dimensional Riemannian orbifold. The $G$-equivariant $p$-widths are defined via sweepouts of $G$-invariant cycles and have the property

$\omega_p^G(M, g) = \omega_p(Q, \widetilde{g}_Q),$

where $\widetilde{g}_Q$ is the rescaled metric on $Q$ induced by the orbit volume function $\vartheta(p)$. Asymptotically, for large $p$,

$$\omega_p^G(M, g) \sim a(1) \left(\mathrm{Vol}(Q, \widetilde{g}_Q)\right)^{1/2} p^{1/2},$$

with $a(1)$ the universal Weyl law constant for 1-dimensional manifolds. This allows direct computation of asymptotic growth rates for the equivariant min-max spectrum and has implications for the density and distribution of minimal $G$-hypersurfaces (Wang, 2023).

6. Generic Density, Applications to Spheres, and Regularity Advances

Almgren–Pitts min-max theory, in the $G$-equivariant cohomogeneity two setting, admits strong generic density results. For any Baire-generic $G$-invariant metric (a $G$-bumpy metric), the union of all minimal $G$-hypersurfaces arising from the equivariant min-max process (as $p$ varies) is dense in $M$, and so are their free boundaries in $\partial M$ if present. The openness and density of such metrics follow from compactness properties combined with parameter-dependent Weyl laws and "metric-squeeze" arguments (Wang, 2023).

Recent works provide comprehensive regularity results for equivariant isotopy minimization problems in cohomogeneity two, completing the theory by establishing full smoothness and multiplicity-one properties for the resulting hypersurfaces in dimensions $4 \leq n+1 \leq 7$ (Ko, 13 Dec 2025). These results extend the toolbox for producing embedded minimal hypersurfaces with rich topology in highly symmetric settings, bypassing gluing PDE techniques in favor of variational and symmetry-based methods.

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Key Papers:

Paper Title	arXiv ID	Main Contribution
Equivariant min-max theory	(Ketover, 2016)	Foundational construction, cohomogeneity-two framework, classical examples, and new infinite families
Generic density of equivariant min-max hypersurfaces	(Wang, 2023)	Weyl law for equivariant $p$-widths, generic density for minimal $G$-hypersurfaces
Regularity of Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces ...	(Ko, 13 Dec 2025)	Regularity and smoothness, construction of large Betti number minimal hypersurfaces on spheres