Generic density of equivariant min-max hypersurfaces

TL;DR

This paper establishes a Weyl law for the $G$-equivariant volume spectrum on certain symmetric manifolds and shows that, generically, min-max minimal $G$-hypersurfaces are dense in the manifold and its boundary.

Contribution

It proves a Weyl asymptotic law for the $G$-equivariant volume spectrum and demonstrates the generic density of min-max minimal $G$-hypersurfaces in the manifold.

Findings

Weyl asymptotic law for $G$-equivariant volume spectrum

Density of min-max minimal $G$-hypersurfaces in $M$

Density of their boundaries in $oundary M$

Abstract

For a compact Riemannian manifold $M^{n+1}$ acted isometrically on by a compact Lie group $G$ with cohomogeneity ${\rm Cohom}(G)\geq 2$, we show the Weyl asymptotic law for the $G$-equivariant volume spectrum. As an application, we show in the $C^\infty_G$-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 $\partial M$.