The existence of G-invariant constant mean curvature 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. Suppose the union of non-principal orbits M \ M reg is a smooth embedded submanifold of M with dimension at most n - 2 . Then for any c ∈ R , we show the existence of a nontrivial, smooth, closed, almost embedded, G -invariant hypersurface Σ n of constant mean curvature c .