The Existence of Embedded $G$-Invariant Minimal Hypersurface

TL;DR

This paper proves the existence of nontrivial embedded minimal hypersurfaces invariant under a group action on certain manifolds, extending minimal surface theory to symmetric spaces with specific group actions.

Contribution

It establishes the existence of embedded G-invariant minimal hypersurfaces for manifolds with group actions of cohomogeneity not equal to 0 or 2, under certain conditions.

Findings

Existence of G-invariant minimal hypersurfaces in specified manifolds.

Hypersurfaces are smooth outside a set of Hausdorff dimension at most n-7.

Applicable to compact orientable Riemannian manifolds with Lie group symmetries.

Abstract

For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.