Generic density of equivariant min-max hypersurfaces

Abstract

For a compact Riemannian manifold $M^{n+1}$ acted isometrically on by a compact Lie group G with cohomogeneity $\mathrm{Cohom}\left(G\right)\ge 2$, we show the Weyl asymptotic law for the G-equivariant volume spectrum. As an application, we show in the $C_G^{\infty }$-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 ∂M.