The Existence of G-Invariant constant mean curvature Hypersurfaces

Abstract

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 without boundary and dim(M \ M reg ) ≤ n − 2. Then for any c ∈ R, we show the existence of a nontrivial, smooth, closed, G-equivariant almost embedded G-invariant hypersurface Σ n of constant mean curvature c.