Min-max theory for $G$-invariant minimal hypersurfaces

TL;DR

This paper extends min-max theory to find $G$-invariant minimal hypersurfaces in certain manifolds with symmetry, providing bounds on widths and demonstrating infinite such hypersurfaces under positive Ricci curvature.

Contribution

It develops a $G$-equivariant min-max framework for minimal hypersurfaces and establishes existence results with bounds, advancing the understanding of symmetric minimal hypersurfaces.

Findings

Existence of nontrivial $G$-invariant minimal hypersurfaces under certain symmetry conditions.

Derived bounds for $(G,p)$-widths analogous to classical results.

Proved infinitely many $G$-invariant minimal hypersurfaces in positively curved manifolds.

Abstract

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. After adapting the Almgren-Pitts min-max theory to a $G$-equivariant version, we show the existence of a nontrivial closed smooth embedded $G$-invariant minimal hypersurface $\Sigma\subset M$ provided that the union of non-principal orbits forms a smooth embedded submanifold of $M$ with dimension at most $n-2$. Moreover, we also build upper bounds as well as lower bounds of $(G,p)$-width which are analogs of the classical conclusions derived by Gromov and Guth. An application of our results combined with the work of Marques-Neves shows the existence of infinitely many $G$-invariant minimal hypersurfaces when ${\rm Ric}_M>0$ and orbits satisfy the same assumption above.