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

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.