Min–Max Theory for G-Invariant Minimal Hypersurfaces

Abstract

In this paper, we consider a closed Riemannian manifold $$M^{n+1}$$ with dimension $$3\le n+1\le 7$$ , 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 non-trivial 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)-widths, 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 $$\mathrm{Ric}_M>0$$ and orbits satisfy the same assumption above.