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 with dimension at most \(n-2\). Then for any \(c\in \mathbb {R}\), we show the existence of a nontrivial, smooth, closed, almost embedded, G-invariant hypersurface \(\Sigma ^n\) of constant mean curvature c.