Min–Max Theory for G-Invariant Minimal Hypersurfaces

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