Min-max theory for free boundary G-invariant minimal hypersurfaces

TL;DR

This paper extends min-max theory for free boundary minimal hypersurfaces to equivariant settings with group actions, proving existence of G-invariant minimal hypersurfaces under certain curvature conditions.

Contribution

It generalizes previous constructions to G-invariant cases, establishing existence and multiplicity results for free boundary minimal hypersurfaces with symmetry.

Findings

Existence of nontrivial G-invariant free boundary minimal hypersurfaces.

Infinitely many properly embedded G-invariant minimal hypersurfaces under positive Ricci curvature.

Extension of min-max theory to equivariant settings with symmetry considerations.

Abstract

Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$ and $\partial M\neq\emptyset$, the free boundary min-max theory built by Martin Man-Chun Li and Xin Zhou shows the existence of a smooth almost properly embedded minimal hypersurface with free boundary in $\partial M$. In this paper, we generalize their constructions into equivariant settings. Specifically, let $G$ be a compact Lie group acting as isometries on $M$ with cohomogeneity at least $3$. Then we show that there exists a nontrivial smooth almost properly embedded $G$-invariant minimal hypersurface with free boundary. Moreover, if the Ricci curvature of $M$ is non-negative and $\partial M$ is strictly convex, then there exist infinitely many properly embedded $G$-invariant minimal hypersurfaces with free boundary.