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

Given a compact Riemannian manifold $M^{n+1}$ with dimension $3\le n+1\le 7$ and $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$, 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 $\mathrm{\partial }M$. 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 $\mathrm{\partial }M$ is strictly convex, then there exist infinitely many properly embedded $G$-invariant minimal hypersurfaces with free boundary.