Infinite existence of equivariant minimal hypersurfaces

Abstract

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show that there are infinitely many $G$-invariant minimal hypersurfaces. Under the assumption that $M$ contains at most a finite number of minimal $G$-hypersurfaces admitting no $G$-invariant unit normal, we further show that each $G$-homology class of $M$ admits infinitely many distinct realizations by embedded minimal $G$-hypersurfaces. The proof relies on a new algorithm that employs multi-stage maximal cuttings. As part of this work, we also established an equivariant min-max theory in manifolds with cylindrical ends.