Infinite existence of equivariant minimal hypersurfaces

TL;DR

This paper proves the existence of infinitely many G-invariant minimal hypersurfaces in certain symmetric manifolds, introducing a new algorithm and an equivariant min-max theory for manifolds with cylindrical ends.

Contribution

It establishes the infinite existence of G-invariant minimal hypersurfaces and develops a novel multi-stage maximal cutting algorithm and equivariant min-max theory.

Findings

Infinitely many G-invariant minimal hypersurfaces exist in the given setting.

Each G-homology class admits infinitely many embedded minimal G-hypersurfaces.

A new algorithm employing multi-stage maximal cuttings was developed for the proof.

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.