The Existence of G-Invariant constant mean curvature Hypersurfaces

Abstract

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\le n+1\le 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits $M\setminus M^{reg}$ is a smooth embedded submanifold of $M$ without boundary and $\dim (M\setminus M^{reg})\le n-2$. Then for any $c\in \mathbb{R}$, we show the existence of a nontrivial, smooth, closed, $G$-equivariant almost embedded $G$-invariant hypersurface ${\Sigma }^n$ of constant mean curvature $c$.