Min–Max Theory for G-Invariant Minimal Hypersurfaces

In this paper, we consider a closed Riemannian manifold M n + 1 with dimension 3 ≤ n + 1 ≤ 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 Σ ⊂ 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 Ric M > 0 and orbits satisfy the same assumption above.