Multiplicity one for equivariant min-max theory in prescribed homology classes

TL;DR

This paper proves a generic multiplicity one result for G-invariant minimal hypersurfaces in equivariant min-max theory and establishes an equivariant min-max framework for hypersurfaces with prescribed mean curvature.

Contribution

It introduces a generic multiplicity one theorem in equivariant min-max theory and develops an equivariant min-max approach for hypersurfaces with prescribed mean curvature.

Findings

Existence of infinitely many G-invariant minimal hypersurfaces in a fixed G-homology class.

Establishment of an equivariant min-max theory for hypersurfaces with prescribed mean curvature.

Proof of a generic multiplicity one property in the equivariant setting.

Abstract

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show a generic multiplicity one theorem in equivariant min-max theory, and show in generic sense that there are infinitely many $G$-invariant minimal hypersurfaces in a fixed $G$-homology class. We also establish an equivariant min-max theory for $G$-invariant hypersurfaces of prescribed mean curvature with $G$-index upper bounds.