The Existence of G-Invariant constant mean curvature Hypersurfaces

TL;DR

This paper proves the existence of smooth, closed, G-invariant hypersurfaces with constant mean curvature in certain symmetric Riemannian manifolds, expanding understanding of geometric structures under symmetry constraints.

Contribution

It establishes the existence of G-invariant constant mean curvature hypersurfaces in manifolds with specific symmetry and orbit structure, under conditions on the group action and orbit types.

Findings

Existence of G-invariant CMC hypersurfaces in specified manifolds.

Construction of such hypersurfaces with prescribed mean curvature.

Extension of known results to higher cohomogeneity actions.

Abstract

In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, 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 ${\rm dim}(M\setminus M^{reg})\leq 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 $c$.