Equivariant Yau conjecture in a prescribed homology class

Let $M$ be a closed Riemannian manifold with a compact Lie group $G$ acting by isometries. A $G$-hypersurface is a $G$-invariant hypersurface, and a $G$-homology class is represented by a fixed $G$-hypersurface $\Sigma_0$ modulo boundaries of open $G$-sets. More precisely, for an open $G$-set $\Omega\subset M$, the relation is taken among mod-$2$ cycles satisfying

$$\Sigma=\Sigma_0+\partial\Omega.$$

Equivariant Yau conjecture. The manifold $M$ contains infinitely many closed embedded minimal $G$-hypersurfaces $\Sigma$ in the given $G$-homology class. This is the equivariant generalization proposed in the paper; the work develops equivariant min-max methods and a generic multiplicity-one theorem, while the full prescribed-class assertion remains open.