Generic density of equivariant min-max hypersurfaces

For a compact Riemannian manifold $M^{n+1}$ acted isometrically on by a compact Lie group $G$ with cohomogeneity $\mathrm{C}\mathrm{o}\mathrm{h}\mathrm{o}\mathrm{m}(G)\ge 2$, we show the Weyl asymptotic law for the $G$-equivariant volume spectrum. As an application, we show in the $C_G^{\mathrm{\infty }}$-generic sense with a certain dimension assumption that the union of min-max minimal $G$-hypersurfaces (with free boundary) is dense in $M$, whose boundaries’ union is also dense in $\mathrm{\partial }M$.